Local Fields and hei Abelian
Ex ensions
Final Deg ee Disse a ion
Deg ee in Ma hema ics
Jo ge Fa i˜na Asa egui
Supe iso :
Gus a o A. Fe n´andez Alcobe
Leioa, 2019-2020
Con en s
P e ace ii
1 P elimina ies 1
1.1 Topological g oups and ields . . . . . . . . . . . . . . . . . . . 1
1.2 P o ini eg oups........................... 3
1.3 In ini e Galois co espondence . . . . . . . . . . . . . . . . . . . 5
1.4 Cha ac e g oups and Pon jagin’s duali y . . . . . . . . . . . . 6
2 Global and Local Fields 10
2.1 Disc e e alua ions . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Comple ion and local ields . . . . . . . . . . . . . . . . . . . . 12
2.3 P ime ideals in ini e ex ensions o global and local ields . . . . 14
3 The B aue G oup 21
3.1 Cen al simple algeb as and he B aue g oup . . . . . . . . . . 21
3.2 G oup cohomology, c ossed p oduc s and cyclic algeb as . . . . 28
3.3 B aue g oup o a local ield . . . . . . . . . . . . . . . . . . . . 31
4 Class Field Theo y 35
4.1 Fini e ields............................. 35
4.2 Local ields ............................. 36
A Sol ed P oblems 44
A.1 P elimina ies ............................ 44
A.2 Global and local ields . . . . . . . . . . . . . . . . . . . . . . . 44
A.3 TheB aue g oup ......................... 46
A.4 Class ield heo y.......................... 51
Bibliog aphy 53
i
P e ace
“In mos sciences one gene a ion ea s
down wha ano he has buil and wha
one has es ablished ano he undoes. In
ma hema ics alone each gene a ion
adds a new s o y o he old s uc u e.”
—He mann Hankel.
Ap il 8 h, 1796. A young Ca l F ield ich Gauss (1777-1855) woke up and
w o e he en y, “Nume o um p imo um non omnes nume os in a ipsos esidua
quad a ica esse posse demons a ione muni um.” 1 o his dia y [Kle03]. He had
jus de ised he i s co ec p oo o he quad a ic ecip oci y law. P e ious
e o s om Fe ma , Eule and Legend e among o he s, had helped o es ablish
his law and pa ial esul s on i s e aci y. Gauss was amazed by he beau y
o his law, which he called Theo ema Au eum (Golden Theo em), and he
managed o p o ide se en mo e di e en p oo s in his li e ime ( h ee o hem
we e published along he i s one in his Disquisi iones A i hme icae in 1801
[Gau01]). Today, mo e han wo hund ed di e en p oo s o his law ha e been
published.
Quad a ic ecip oci y shows an imp essi e simme y ha allows o de e -
mine i a p ime pis a squa e modulo a p ime q, by looking whe he qis a
squa e modulo p. A na u al gene aliza ion is o ind highe ecip oci y laws,
i.e. cubic, cua ic, quin ic, e c. This leads di ec ly o ex ending he ield o a-
ionals o mo e elabo a ed numbe ields. E ns Kumme ’s (1810-1893) ideal
numbe s, a p ecu so o ideals la e de eloped by Richa d Dedekind (1831-
1916), came o exis ence in sea ch o hese highe ecip oci y laws.
In 1900, Da id Hilbe (1862-1943) made up a lis o wen y h ee p oblems
conce ning some o he mos ele an unsol ed ques ions o his ime. Among
hose p oblems, P oblem 9 h deals wi h gene al ecip oci y laws [Hil00]:
1P ime numbe s below (modulo) all numbe s may no be quad a ic esidues, possesses
a ough p oo .
ii
“Fü einen beliebigen Zahlkö pe soll das Recip oci ä sgese z de l en
Po enz es e bewiesen we den, wenn leine unge ade P imzahl bedeu e und
e ne , wenn leine Po enz on 2 ode eine Po enz eine unge aden P imzahl
is . Die Au s ellung des Gese zes, sowie die wesen lichen Hül smi el zum Be-
weise desselben we den sich, wie ich glaube, e geben, wenn man die on mi
en wickel e Theo ie des Kö pe s de l en Einhei swu zeln 1) und meine The-
o ie 2) des ela i -quad a ischen Kö pe s in gehö ige Weise e allgemeine .”2
Theo y o ideals was u he de eloped and Emil A in (1898-1962) p o-
ided he i s p oo o his gene al ecip oci y law in a se ies o pape s a ound
1927, which implies all known ecip oci y laws. This was deduced p o ing he
main heo em o global class ield heo y, which desc ibes abelian ex ensions
o a global ield in e ms o i s a i hme ic in insic p ope ies. A good accoun
o his p ocedu e is gi en in [Lan94] o example.
La e , he app oach u ned abou o he local-global p inciple. The local
e sion o class ield heo y, i.e. o local ields, was i s p o ed by de e mining
he B aue g oup o a local ield (see Chap e 3), and he global e sion was
ob ained by conside ing all p imes a once ia ideles and adeles. This p oce-
du e u ned ou o be be e unde s ood in he language o g oup cohomology,
and his is he way i is cu en ly p esen ed [AT09].
Nowadays, gene alisa ions o non abelian ex ensions a e being de eloped
ex ending he heo y o abelian ex ensions. In his wo k we shall seek o s udy
local class ield heo y.
Le us show he eade an in ui i e mo i a ion o class ield heo y. Gi en
a ield Kand a ini e Galois ex ension L/K, he main heo em o Galois he-
o y desc ibes he in e media e ield ex ensions in e ms o he subg oups o
he Galois g oup. Howe e , his p ocedu e needs he ini e ex ension ield L
o be ixed i s . We may wonde i we can gi e a desc ip ion o all he ini e
ex ensions o a gi en ield. This is no an easy ea o accomplish wi h all
gene ali y3, bu i is easie i we es ic ou a en ion o ini e abelian ex en-
sions. In he case whe e he base ield is ei he he ield o complex numbe s
o he eals, his desc ip ion is i ial since Cis algeb aically closed and Rhas
a unique abelian ex ension, namely C.
2Fo any numbe ield he ecip oci y law o l h esidues should be p o ed, when lis
an odd p ime and when lis a powe o 2 o o an odd p ime. The lis o laws, as well
as complemen a ies o he p oo i sel should be ob ained, in my opinion, om my well-
de eloped l h cyclo omic ield heo y 1) and an app opia e gene aliza ion o my heo y 2)
o ela i e-quad a ic ields.
3The e a e s ill open p oblems conce ning he Galois g oup Gal(Qal/Q), such as whe he
each ini e g oup occu s as a quo ien o i [Mil20].
Le Eand Fbe wo ini e abelian ex ensions o K. Then, he composi e
EF is also Galois. Wha is mo e, Gal(EF/K)is isomo phic o a subg oup o
he ca esian p oduc Gal(E/K)×Gal(F/K), which is abelian; hus, EF/K
is abelian oo.
Bu , wha can be said abou he composi e o a coun able numbe o
abelian ex ensions? Le us see a mo i a ing example. Le he base ield be he
ield o a ionals. Then, i is known om he cou se in Algeb aic Equa ions
ha o each na u al numbe n, he n h cyclo omic ex ension is abelian. Since
he deg ee o he n h cyclo omic ex ension is p ecisely ϕ(n), hese deg ees a e
no bounded and he composi e o hem all is an in ini e ex ension. We lea e
o he eade he de ails on why his composi e is an abelian Galois ex ension
o Q, as a p epa a ion o he ollowing explana ions which shall gene alize
his example o a gene al ield ex ension L/K.
Le L/K be a gene al Galois in ini e ield ex ension and de ine he se
L:= {E:E ini e Galois in e media e ex ension o L/K}.
No e ha o any pai E, F ∈ L hei composi e EF ∈ L. This makes Lin o
a di ec ed pose wi h espec o he inclusion. Also, no e ha o any pai
E, F ∈ L he na u al inclusions ϕEF :E→Fli he elemen s in E o F,
whene e E⊆F. This makes Lin o a di ec sys em o e i sel . No e ha he
same na u al li ings ϕE:E→Lexis o each E∈ L. Wha is mo e, hese
li ings a e compa ible wi h he ones in he di ec sys em, i.e. , ϕE=ϕEF ϕF
whene e E⊆F. Then, by he uni e sal p ope y o he di ec limi (see
Chap e 1), Lcan be ega ded as he di ec limi lim
−→E=SE. Then, any
elemen in Llies in a ini e Galois ex ension o K,E, and he e is a e y
na u al way o desc ibe he K-au omo phisms o L: as cohe en uples (σE)E.
An expec ed, bu which he eade should p o e, p ope y o an in ini e
Galois ex ension L/K is ha i s Galois g oup is in ini e (hin : show he de-
g ees o in e media e ields a e no bounded).
We could expec he ini e Galois co espondence o gene alize immedi-
a ely o he in ini e case. Sadly, his is no he case. In pa icula , no all
ini e index subg oups o Gal(L/K)need co espond o ini e in e media e
ex ensions, i.e. hey may no be o he o m Gal(L/E)(see P oblem A.1).
Howe e , his may be ixed by endowing he Galois g oup wi h a special
opology whe e hese subg oups a e p ecisely open, u ning Gal(L/K)in o
a opological g oup. Le Gdeno e he Galois g oup o he ex ension L/K.
Gi en an in e media e ield EGalois o e K, he e is a na u al p ojec ion
om G o Gal(E/K)by es ic ion o au omo phisms. Also, o in e media e
Galois ield ex ensions K⊆E⊆F⊆L, he e is a na u al composi ion o
es ic ions G→Gal(F/K)→Gal(E/K)which is no mo e han he usual
es ic ion G→Gal(E/K). Wha is mo e, o any pai o in e media e Galois
ield ex ensions K⊆E, F ⊆L, he e exis s ano he in e media e Galois ex-
ensions ield, namely he composi e EF , con aining bo h Eand Fand whose
co esponding subg oup Gal(L/EF)is p ecisely he in e sec ion o bo h sub-
g oups Gal(L/E)and Gal(L/F). Then, hese subg oups o m a il e o no mal
subg oups. This il e can be used o gi e a opology o G, which is called he
K ull opology. Also, i gi es us as he da a o an in e se sys em, whe e he
objec s a e he ini e Galois g oups Gal(E/K)endowed wi h he disc e e opol-
ogy and he connec ion homomo phisms a e he abo e es ic ions. Thus, we
may iden i y Gwi h he in e se limi lim
←−Gal(E/K) ia he a o emen ioned
p ojec ions and he uni e sal p ope y o he in e se limi (see Chap e 1).
We shall see in Chap e 1 ha bo h cons uc ions coincide up o isomo phism
and endow Gwi h he same opology.
Now, we es ic ou a en ion o abelian ex ensions o a ield K. Applying
p e ious easoning wi h he added condi ion he ex ensions a e abelian, i.e.
le ing L:= {E:E/K is ini e abelian}, he di ec limi exis s and i con-
ains all he abelian ex ensions o K. This di ec limi is called he maximal
abelian ex ension o Kand deno ed by Kab. Then, by he in ini e Galois co -
espondence (see Chap e 1), s udying he ini e abelian ex ensions o Kis
equi alen o s udying open subg oups o Gal(Kab/K). The d awback is we
ha e de ined hese open subg oups as he ones coming om ini e in e media e
ex ensions and a p io i we ha e no way o dis inguish among hem. The spe-
cial case whe e Kis a ini e ield is e y well known and we shall s udy i i s
o come up wi h a mo i a ion o o he ields. And his is p ecisely he objec-
i e o class ield heo y: s udying hese open subg oups ia an easie - o-s udy
g oup. Wha is mo e, we shall show his easie - o-s udy g oup is ela ed o he
a i hme ic o he base g oup K, making he da a o abelian ini e ex ensions
o Kin insic o K, which, in my humble opinion, is a esul o an as onishing
beau y.
We aim o p o ide a comple e p oo o local class ield heo y. To ul ill
his goal, we ake a mo e algeb aic app oach, a he han he mode n coho-
mological pe spec i e. We ollow he ou line in [KKS11] and ill in he de ails
om o he sou ces and ou sel es, ying o make he p oo as sho and as
easy as possible since mos ex s ake a lo longe o p o ide a ull p oo o his
heo em. Ine i ably we will be missing many in e es ing concep s and heo ies
ha a ise in ou discussion, which could ake a whole book by hemsel es. We
ha e no space ei he o deduce he global e sion o class ield heo y, which
is a beau i ul applica ion o he local-global p inciple.
In he i s chap e we in oduce some p elimina y concep s such as opo-
logical g oups, p o ini e g oups, cha ac e g oups and Pon jagin’s duali y,
which shall appea in he es o he wo k. We ollow mos ly [RZ10].
In he second chap e , we p esen basic ac s abou local and global ields
and desc ibe ami ica ion o p ime ideals in hei abelian ex ensions. Due o
lack o space, concep s such as di e en s o disc iminan s a e no e en men-
ioned. We p o ide a sho in oduc ion o opological g oups and ields, in
o de o s a e Pon jagin’s duali y, which is undamen al o he p oo o lo-
cal class ield heo y. We ollow mos ly [FV02] and [SG80] o s a e and p o e
esul s on comple ions. Fo he es o he chap e we ollow [KKS11].
The hi d chap e is de o ed o de e mining he B aue g oup o local
ields. Fo ha , we in oduce he heo y o di ision algeb as, simple cen al
algeb as, c ossed p oduc s and cyclic algeb as, as well as a li le in oduc ion
o classic g oup cohomology. We ollow mos ly [Jac85] o he heo y o simple
cen al algeb as and [KKS11] o he de e mina ion o he B aue g oup o a
local ield. The cohomological in oduc ion is aken om [Mo 96].
The las chap e is whe e a p oo o local class ield heo y is gi en. We
i s show he ini e ield case as a p ecu so o local ields.Then, we p o e
local class ield heo y wi h all he ools om p e ious chap e s. We ollow
and comple e he p oo s in [KKS11].
We ha e summa ized some mino esul s ha had no place on he main
low o he ex in an appendix as sol ed exe cises, jus o comple eness. Mos
p oblems a e aken om he same sou ces as he main ex , bu some a e aken
om o he sou ces. Fo example, Dedekind’s Independence Theo em has been
aken om [Jac09], which is no used o he main ex . A second appendix
con ains he essen ials o he heo y o in e se limi s, di ec limi s and p o i-
ni e g oups, jus in case he eade is un amilia wi h hese no ions, allowing
he eade o ollow he explana ions in his wo k lawlessly.
The eade is no assumed o ha e any p io knowledge apa om wha
is augh in his deg ee.
Las ly, a li le no a ion issue. Map composi ion will be w i en mul iplica-
i ely om le o igh , i.e. o maps and g, hei composi ion (also called
p oduc ) will be w i en g whe e i s we apply and hen g.
Commu a i e ings will be assumed o ha e an iden i y, see [Poo14] o a
sho nice discussion on his.
1.2. P o ini e g oups
may cons uc he in e se limi as he closed subse o he ca esian p oduc
Qi∈I Xi o med by all cohe en uples (a he le el o elemen s), i.e. he uples
(xi)i∈I such ha ϕij(xi) = xjwhene e ji. I is le o he eade o p o e
his cons uc ion sa is ies he uni e sal p ope y o he in e se limi [RZ10].
This educes many imes p ope ies o in e se limi s o objec s in C o p op-
e ies o he objec s in C. In pa icula , i he objec s a e ini e g oups, hen
he esul ing in e se limi will beha e simila ly o a ini e g oup and many
p ope ies will be deduced by educing i o he ini e case.
Again, he de ini ion o he di ec limi is he dual no ion o he in e se
limi and i is le o he eade he de ails o i s de ini ion (hin : e e se all
a ows in he de ini ion o he in e se limi ). In ui i ely, he di ec limi is a
union. In he ca ego y o abelian opological g oups X= lim
−→Xi=Siϕi(X)
and X=SiXii he p ojec ions a e on o [RZ10]. In he in oduc ion, we
ha e seen how a ield ex ension may be seen as a di ec limi , and we shall
see how aking i s g oup o au omo phisms dualizes i u ning in o an in e se
limi and inally applying he hom unc o gi es us a di ec limi again. This
scheme applies in o he si ua ions oo and i will be i al o us.
We will be wo king wi h ini e g oups, which can be endowed wi h he
disc e e opology o u n hem in o opological g oups. In his con ex , he
in e se limi is compac , Hausdo and o ally disconnec ed [RZ10] and i is
called a p o ini e g oup.
Gi en a g oup Gwe may de ine he di ec ed se o no mal subg oups
N:= {N≤ G:G/N ini e}. Then, we de ine he p o ini e comple ion o
G, deno ed by b
Gas he in e se limi lim
←−G/N whe e N uns o e N. Then,
Gis na u ally embedded in o i s p o ini e comple ion by he ob ious map
g7→ (gN)N.
The opological closu e o a subg oup o a p o ini e g oup can be ob ained
as ollows.
Lemma 1.3. Le Gbe a p o ini e g oup and Ha subg oup o G. Then, he
opological closu e o Hin Gcan be ob ained as
H=
N
HN ∼
=lim
←−HN/N,
whe e N uns o e all open no mal subg oups in G.
We shall see in he nex sec ion ha we a e in e es ed in he open sub-
g oups o he Galois g oup Gal(Kab/K). Class ield heo y will be based upon
an easie - o-s udy g oup whose p o ini e comple ion is p ecisely his Galois
g oup (up o isomo phism). Then, he ollowing p oposi ion [RZ10] is i al o
us o ansla e he Galois co espondence o his easie - o-s udy g oup.
4
Chap e 1. P elimina ies
P oposi ion 1.4. Le Gbe a esidually ini e g oup, i.e. he in e sec ion
o all i s no mal subg oups o ini e index is i ial. Then, he e is a 1- o-1
co espondence be ween he open subg oups in Gand he open subg oups in i s
p o ini e comple ion b
Ggi en by H7→ Hand K7→ K∩G espec i ely.
1.3 In ini e Galois co espondence
We s a e wi hou p oo he in ini e e sion o he main heo em o Galois
heo y [Mo 96] e en i we jus need he asse ion on ini e ex ensions.
Theo em 1.5. Le L/K be an in ini e Galois ex ension and Gi s Galois
g oup endowed wi h he K ull opology. Then, he e is a one- o-one inclusion
e e sing co espondence be weeen he closed subg oups o Gand he in e -
media e ex ensions o L/K. Wha is mo e, he in e media e ex ension Eis
no mal i and only i i s co esponding closed subg oup N:= Gal(L/E)is
no mal in G, in which case G/N ∼
=Gal(E/K)as opological g oups. Also,
i we es ic o ini e ex ensions, his co espondence is one- o-one be ween
ini e in e media e ex ensions and open subg oups o G.
We shall see he in e se limi lim
←−Gal(E/K)is isomo phic o Gwi h he
K ull opology as opological g oups. Since he na u al p ojec ions G→
Gal(E/K)a e compa ible wi h he connec ion homomo phisms o being he
usual es ic ions as abo e, by he uni e sal p ope y o he in e se limi , he e
exis s a unique g oup homomo phism Φ : G→lim
←−Gal(E/K)compa ible wi h
he co esponding p ojec ions. By cons uc ion, his homomo phism is p e-
cisely he one gi en by σ7→ (σ|E)E. We shall see i is an isomo phism. The
ke nel is i ial since he only au omo phism ha es ic s o he iden i y in
all ini e in e media e ields is he iden i y. Fo ha , ecall L= lim
−→E=SEE
whe e E uns h ough all he ini e in e media e ex ensions in L/K. Then, o
each s∈L,s∈E o some Eand since σE(s) = s o all Eand all s∈L,σ
is he iden i y as wan ed.
To show i is on o we shall see ha each in ini e cohe en uple (σE)Eli s
o an au omo phism in Gal(L/K)whe e cohe ence means ha o in e medi-
a e ields E⊆Fand a uple (σE)E, he componen σF es ic s o σEin E.
Then, since each s∈Lis con ained in some ini e Galois ex ension o K,E, we
may de ine he au omo phism σ∈Gas s7→ σ(s) := σE(s). I is well de ined
since i sis in ano he in e media e Galois ex ension ield F,σE(s) = σF(s).
To see ha 1, no e ha K(s)⊆E, F, and o being cohe en , he es ic ions
o σEand σF o K(s)coincide; hus, hei image on s oo. No e his au o-
mo phism is ac ually an au omo phism and ixes K; hus, σ∈G. Clea ly, all
au omo phisms in Gcan be ob ained in his ashion, since hei es ic ion o
1We could ha e used EF ins ead o K(s) oo and apply cohe ence he e.
5
1.4. Cha ac e g oups and Pon jagin’s duali y
each in e media e ini e ex ension o m cohe en uples (σE)E.
Then, we ob ain an isomo phism o g oups be ween Gand he in e se limi
lim
←−Gal(E/K). We need o make i in o a homeomo phism. We shall copy he
opology in he in e se limi o G h ough he abo e isomo phism. We know
ha a undamen al sys em o open neighbo hoods in lim
←−Gal(E/K)is gi en
by he ke nels o he p ojec ion homomo phisms (Lemma 2.1.1 in [RZ10]).
Thus, since hese a e usual es ic ions/p ojec ions o Gin o Gal(E/K)∼
=
G/Gal(L/E) o in e media e ini e Galois ex ension ields E he ke nels a e
p ecisely he no mal subg oups Gal(L/E). Then, {Gal(L/E)}E o ms a un-
damen al sys em o open neighbo hoods in G, which is p ecisely he way we
de ined he K ull opology.
1.4 Cha ac e g oups and Pon jagin’s duali y
The cha ac e g oup o dual o a opological g oup, G∗, is he g oup o con-
inuous g oup homomo phisms om G o T, i.e. G∗= homcon (G, T), whe e
Tis he mul iplica i e subg oup o complex numbe s o uni no m.
We will be dealing wi h cha ac e g oups h oughou he p oo o local
class ield heo y. Then, we shall ix some special no a ion and show a cou-
ple o esul s. Fo a ield Kwe deno e he cha ac e g oup o Gal(Kab/K)
by X(K). Since we es ic o con inuous homomo phisms, we see ha he
p eimage o any a om in Tis open, since Tis gi en he disc e e opology.
Bu open in compac opological g oups implies ini e index. In pa icula ,
he ke nel is o ini e index, i.e. he image o he gi en homomo phism is o
ini e o de . Thus, he image g oups o hese homomo phisms a e mapped o
Q/Z ia he usual isomo phism e2πix 7→ x+Z. Then, o a p o ini e g oup
G∗= homcon (G, Q/Z). Also, i we de ine addi ion o homomo phisms ia
addi ion o hei images, X(K)can be seen as an addi i e g oup and since
o a homomo phism ϕ,o(ϕ) = lcm(o(ϕ(g)))g∈G, he addi i e o de o any
homomo phism is ini e and X(K)is o sion.
All cha ac e s o an abelian p o ini e g oup a ise as cha ac es o a ini e
abelian g oup. No e ha o any χ∈G∗,ke χis no mal in Gand open oo
o being he p eimage o 0 and Q/Zbeing disc e e. Then, we can ega d χas
he na u al composi ion G→G/ ke χ→Q/Z.
A use ul p ope y o cha ac e s is ha gi en a g oup homomo phism
ϕ:G→Hwhe e His abelian and ini e, i will be on o i and only i
he only cha ac e annihila ing he image o ϕis he i ial cha ac e , i.e. i
AnnH∗(ϕ(G)) = 0. No e he only i pa is i ial. Fo he i pa assume by
6
Chap e 1. P elimina ies
con adic ion ha he e exis s an elemen h∈Hsuch ha h /∈Im χ. We shall
ind a non i ial cha ac e χ∈H∗such ha χ(ϕ(G))=0. Fo ha conside he
non- i ial bu ini e quo ien H/ϕ(G). Since he quo ien is ini e and abelian,
Tχ∈(H/ϕ(G))∗ke χ= 0, he e is a leas one cha ac e χin (H/ϕ(G))∗which
is non- i ial and we may li i o a cha ac e in o Hby χ0:= πχ, ge ing
a non i ial cha ac e o Hannhila ing he image o Gby ϕ, a i ing o a
con adic ion and p o ing he desi ed p ope y.
Pu ing e e y hing oge he , le L/K be an in ini e Galois ex ension. Le
us ecall he de ini ion o he se Land how na u al i is o cons uc he
di ec limi o his di ec sys em, lim
−→E=SE=L, which is no mo e han he
in ini e Galois ex ension L. Now, we shall conside he Galois g oups o each
ini e Galois ex ension o Kin L. These g oups Gal(E/K) oge he wi h he
usual es ic ions ϕEF : Gal(F/K)→Gal(E/K)gi e us he da a o an in e se
sys em o e he same di ec ed pose L. Now, i is na u al again o conside
he in e se limi lim
←−Gal(E/K)∼
=Gal(L/K). No e ha Galois co espondence
being inclusion e e sing u ns inclusions in o es ic ions, dualizing he con-
s uc ion, u ning a di ec limi in o an in e se limi . This had been shown so
a in ou discussion.
Now, conside he dual o each ini e Galois g oup, i.e. he homomo -
phisms om Gal(E/K) o Q/Z. No e ha a cha ac e χE: Gal(E/K)→Q/Z
li s o a cha ac e χF: Gal(F/K)→Q/Zwhene e E⊆Fand χF(σ) :=
χE(σ|E) o each σ∈Gal(F/K). In ui i ely, we a e plugging he bigge g oup
Gal(F/K)in he le ia a es ic ion. This allows us o li he cha ac e s o
he smalle g oup Gal(E/K) o he bigge g oup Gal(F/K). This gi es us he
da a o a di ec sys em o e he same di ec ed pose L. Na u ally, we build
he di ec limi lim
−→Gal(E/K)∗∼
=Gal(L/K)∗. To ob ain ha isomo phism,
ecall he image o each cha ac e is ini e. Hence, any cha ac e can be ob-
ained in his li ing ashion. To see ha , no e his ke nel is a subg oup o
he o m Gal(L/E)whe e Eis a ini e Galois ex ension o K o he ke nel
being open. Then, his cha ac e can be ob ained om a cha ac e o he ini e
Galois g oup Gal(E/K)by plugging in he le Gal(L/K) h ough he usual
es ic ion. Then he isomo phism ollows om he uni e sal p ope y o he
di ec limi , since he li ings a e compa ible wi h he connec ion homomo -
phisms as hey a e also usual inclusions.
Las ly, we wan o build he bidual Gal(L/K)∗∗. Fo ha , conside he
bidual o each ini e Galois g oup, Gal(E/K)∗∗. These o m an in e se sys em
o e he same di ec ed pose L. Jus no e a cha ac e χE: Gal(E/K)∗→Q/Z
es ic s o a cha ac e χF: Gal(F/K)∗→Q/Zsince he bidual o a ini e
g oup is known o be isomo phic o he o iginal ini e g oup h ough he
e alua ion homomo phism and we may de ine his es ic ion h ough hese
isomo phisms and he usual es ic ion in he Galois g oups. Then, we conside
7
1.4. Cha ac e g oups and Pon jagin’s duali y
he in e se limi lim
←−Gal(E/K)∗∗ ∼
=Gal(L/K)∗∗. No e hese e alua ion iso-
mo phisms a e compa ible wi h he es ic ions by de ini ion; hus, we ob ain
he e alua ion isomo phism o in e se limi s Gal(L/K)∗∗ ∼
=lim
←−Gal(E/K)∗∗ ∼
=
lim
←−Gal(E/K)∼
=Gal(L/K). This is known wi h mo e gene ali y o p o ini e
g oups as Pon jagin’s duali y and e en in mo e gene ali y o locally compac
abelian g oups.
Theo em 1.6 (Pon jagin’s duali y). Gi en a locally compac abelian
g oup Gand i s cha ac e g oup G∗, he cha ac e g oup o G∗and he g oup
Ga e na u ally isomo phic, whe e his isomo phism is gi en by he e alua ion
map.
The in e es ed eade is ad ised o check [Pon46] o a ull p oo o Pon-
jagin’s duali y o locally compac abelian g oups and [RZ10] o p o ini e
g oups.
Rema k. The eade may be wonde ing why we a e using es ic ions ins ead
o li ings when conside ing he Galois g oups. In he case o cha ac e s we
a e able o do hese li ings because we a e conside ing jus homomo phisms,
no au omo phisms. I we y o li an au omo phism in his ashion i will
ha e a non- i ial ke nel and hus i will no be injec i e, i will no be e en
a homomo phism since ield homomo phisms a e injec i e. This shows why in
his case i is na u al o conside es ic ions and he e o e he in e se limi
whils wi h cha ac e s i is mo e na u al o li hem and conside he di ec
limi .
Le us analyze he case K=Fq. Then, all ini e in e media e ields a e
known o us o be cyclic and we ha e a mo e p ecise desc ip ion o L=
{Fqn:n∈N}. Then, Fab
q=Fsep
q= lim
−→Fqn=SFqnand he Galois g oup
Gal(Fab
q/Fq)∼
=lim
←−Gal(Fqn/Fq)∼
=lim
←−Z/nZ=ˆ
Z. Finally, since Q/Zis dis-
c e e and each ini e g oup in Lis disc e e, o know all con inuous homomo -
phisms Gal(Fqn/Fq)→Q/Zis jus o know all such g oup homomo phisms.
Fo being he base g oup ini e and cyclic, i is enough o p o ide he image o
he F obenius au omo phism, and choose i o be an elemen in Q/Zo o de
di iso he o de o he g oup, i.e. any elemen in h1/n +Ziwhe e nis his
o de . Then, X(Fq)∼
=lim
−→Gal(Fqn/Fq)∗∼
=lim
−→h1/n+Zi=Sh1/n+Zi=Q/Z.
Thus, X(Fq)∼
=Q/Zwhe e his isomo phism is gi en by χ=χn7→ k/n +Z,
whe e k/n is he image o he n h F obenius au omo phism unde χn=πnχ.
In o he wo ds, his isomo phism is ob ained mapping each cha ac e χ o i s
image in he F obenius au omo phism o Gal(Fab
q/Fq). No e his a gumen is
alid o any ini e ield, in pa icula o Fq . Then, X(Fq)∼
=X(Fq ). Bu
we know explici ly a e y na u al homomo phism be ween hese wo cha ac-
e g oups (which is no an isomo phism, cau ion!). Since he Galois g oups
Gal(Fqn/Fq)a e cyclic, whene e di ides n he e is a unique subg oup o
index , namely Gal(Fqn/Fq ). Wha is mo e, an h powe o a F obenius
8
Chap e 1. P elimina ies
au omo phism in Gal(Fqn/Fq)is a F obenius au omo phism in Gal(Fqn/Fq ).
Thus, gi en a homomo phism Gal(Fqn/Fq)→Q/Z,σ7→ k/n +Zwe ob ain a
homomo phism Gal(Fqn/Fq )by he mul iplica ion-by- map in Q/Zand he
usual powe -by- map in he Galois g oups, σ7→ k/n +Z
Gal(Fqn/Fq)Q/Z
Gal(Fqn/Fq )Q/Z
es ic ion mul iplica ion by
whene e ndi ides . Mul iplica ion-by- is an epimo phism in each e m
o he di ec ed se , and i we conside he cha ac e g oups and he na u al
isomo phisms mapping a cha ac e o i s image in he co esponding F obenius
au omo phisms, we ge he ollowing commu a i e diag am which we shall use
in he p oo o local class ield heo y.
Lemma 1.7. The diag am
X(Fq)Q/Z
X(Fq )Q/Z
es ic ion mul iplica ion by
is commu a i e.
9
Chap e 2
Global and Local Fields
The i s example o a ield seen in an elemen a y algeb a cou se is usually he
ield o a ional numbe s, Q. Fini e ex ensions o Qa e called numbe ields,
and hey a e he main objec o s udy in algeb aic numbe heo y. Along wi h
numbe ields (and ini e ields), unc ion ields a e he bes known examples o
ields. O special in e es in algeb aic geome y a e ini e ex ensions o Fq(T),
whe e Fq(T)is he ield o a ional unc ions in one a iable wi h coe icien s
in Fq. We call he la e ields global unc ion ields. These wo ypes o ields
may look a he di e en a i s glance, bu hey sha e many p ope ies. This
analogy be ween bo h o hem mo i a es he de ini ion o a global ield as ei-
he a numbe ield o a global unc ion ield. Pu suing hese analogies has
been shown ui ul o bo h algeb aic geome y and numbe heo y. In his
chap e we de elop he basic heo y o bo h global and local ields ha is used
in subsequen chap e s.
We will use he ollowing equi alen [Mil17] de ini ions o a Dedekind do-
main h oughou he chap e .
De ini ion 2.1 (Dedekind domain). An in eg al domain Ais said o be a
Dedekind domain i i is a ield o any o he ollowing equi alen condi ions
is sa is ied.
1. Any p ope ideal ain A ac o s uniquely in o a p oduc o p ime ideals.
2. Ais noe he ian, in eg ally closed and e e y nonze o p ime ideal is max-
imal.
2.1 Disc e e alua ions
De ini ion 2.2 (Disc e e alua ion). Le Kbe a ield. Le ν:K×→
Zbe a non- i ial su jec i e g oup homomo phism sa is ying he addi ional
condi ion
10
Chap e 2. Global and Local Fields
(i) ν(a+b)≥min(ν(a), ν(b)),∀a, b ∈K×,
and se ν(0) = ∞ o ex end ν o he en i e ield K. Then, νis said o be a
disc e e alua ion o K. A ield wi h a disc e e alua ion is called a disc e e
alua ion ield.
The i s example o a disc e e alua ion o a ield is he map o dp:Q×→Z
de ined as ollows o a a ional p ime p. Fo any a ional numbe a, le us
w i e i as an i educible ac ion in he o m
a=pna0
a1
, n ∈Zand p-a0, a1.
We de ine o dp(a) = nand se o dp(0) = ∞. I is easy o check ha o dp
is a disc e e alua ion o Q. This map is called he p-adic alua ion.
Simila ly, o a Dedekind domain Aand i s ield o ac ions K, we de-
ine he p-adic alua ion,o dp:K×→Z o a nonze o p ime ideal po A,
by w i ing he ac ional ideal (a) o any a∈K×as he unique p oduc o
p ime ideals (a) = pnQqni
iwi h n, ni∈Zand qi6=p o all i, and de ining
o dp(a) = n. No e ha we se o dp(0) = ∞as be o e.
Le Kbe a disc e e alua ion ield o ν. Then, i is immedia e om he
de ini ion o a disc e e alua ion ha he se Oν={a∈K:ν(a)≥0} o ms
a sub ing o Kand i is called he alua ion ing o K.
P oposi ion 2.3. Le Kbe a ield and νa alua ion o K.
(i) Le Oνbe he alua ion ing o Kwi h espec o ν. Then, Oνis a
p incipal ideal domain and hus a Dedekind domain. The only nonze o
p ime (and maximal) ideal o Oνis
p={a∈K:ν(a)≥1},
and νcoincides wi h o dp. Any elemen ain Ksuch ha ν(a)=1
gene a es p; any ideal o Oνis o he o m (an) = {b∈K:ν(b)≥n}
o such an elemen aand n∈N, and any ac ional ideal o Oνo he
o m (an) = {b∈K:ν(b)≥n} o same aand n∈Z. The g oup o
uni s o Oνis exac ly he se o he elemen s wi h null alua ion, i.e.
O×
ν={a∈K:ν(a)=0}.
(ii) Con e sely, le Abe a Dedekind domain wi h a unique non-ze o p ime
ideal p. Then, Acoincides wi h he alua ion ing o he disc e e alu-
a ion o dp.
(iii) Gi en an in eg al domain A, he ollowing condi ions a e all equi alen .
11
2.2. Comple ion and local ields
(a) Ais he alua ion ing o a disc e e alua ion o i s ield o ac-
ions.
(b) Ais a p incipal ideal domain wi h a unique nonze o p ime ideal.
(c) Ais a Dedekind domain wi h a unique nonze o p ime ideal.
An in eg al domain Asa is ying any o he las h ee equi alen condi-
ions is called a disc e e alua ion ing.
P oo . Fi s no e ha an elemen c∈ Oνo null alua ion is a uni in he al-
ua ion ing since 1 = cc−1in Kimplies aking alua ions, 0 = ν(c) + ν(c−1);
hus, ν(c−1)=0and c∈ O×
ν. Simila ly, i c∈ O×
ν, hen ν(c)=0. Le a
be an elemen o Ksuch ha ν(a) = 1. Now ix a nonze o ideal ao Oν.
Le n= min{ν(b) : b∈a}. Then, by he de ini ion o an ideal, a⊆b:=
{b∈K:ν(b)≥n} ⊇ (an). To p o e hese inclusions a e ac ually equali-
ies, le i s b∈b. Then, b=ancwhe e c=a−nb∈K. Taking alua ions,
ν(c) = ν(a−n) + ν(b)≥0. Thus, c∈ Oνp o ing b∈(an). Le now b∈a
such ha ν(b) = n. Then, b=anc o some c∈Kand aking alua ions,
ν(b) = ν(c) + ν(an). Thus, ν(c) = 0 and c∈ O×
ν, so an=c−1b∈age ing
bo h equali ies. The o he assse ions in (i) a e s aigh o wa d o check now.
Fo (ii) jus no e Ais i ially con ained in he alua ion ing. Fo he
e e se inclusion no e ha o an elemen a∈ Oo dp,(a) = pnwi h n≥0;
hus, a∈A.
Now, (iii) ollows om (i) and (ii).
A gene a o πo he unique maximal ideal po a disc e e alua ion ing
is called a uni o mize o a p ime elemen o Oνo K. The na u al quo ien
ield Oν/pis called he esidue ield o Oν.
We shall see how p ime ideals o igin embeddings o global ields in o wha
we call local ields. Fo ha , we need o a ach a special opology o hese
ields.
2.2 Comple ion and local ields
F om a disc e e alua ion ν, we can ob ain a me ic. Le cbe a eal numbe
such ha 1<c<∞. Then, i is easily checked ha he map dν:K×K→R
de ined as dν(x, y) := c−ν(x−y) o x6=yand dν(x, y) := 0 o x=y, de ines
a me ic in K. This me ic induces a Hausdo opology whe e Vn,a ={b∈
K:ν(b−a)≥n}can be aken as a undamen al sys em o open (and closed)
neighbo hoods o each poin ain K.
12
Chap e 2. Global and Local Fields
As wi h espec o he usual me ic in Q, a Cauchy sequence in Kwi h
espec o dνmay no con e ge in K. Thus, we may comple e Kwi h espec
o his me ic o make all Cauchy sequences con e ge.
Lemma 2.4. Le Abe he se o all Cauchy sequences in K. Then, Ais a
ing wi h espec o componen wise addi ion and mul iplica ion. The se o
all Cauchy sequences con e gen o 0 o m a maximal ideal o A,m. The
ield A/mis a disc e e alua ion ield wi h disc e e alua ion ˆνde ined as
ˆν((an) + m) = lim ν(an).
P oo . P o ing Ais a ing is s aigh o wa d; hus, i is le o he eade .
To p o e mis maximal, le m0be an ideal s ic ly con aining m. We shall p o e
m0=A. Take a Cauchy sequence (an)in m0 m. Then, he e exis s a posi i e
in ege n0such ha an6= 0 o n≥n0. Le (bn)be a sequence such ha o
n≥n0,bn=a−1
n. Then, (bn)is clea ly Cauchy and (an)(bn) + m= (1) + m.
Thus, since m0is an ideal, (1) is in m0and m0=Aas wan ed. The ac ˆν
is a disc e e alua ion o A/m ollows now om he p ope ies o he usual
limi .
A disc e e alua ion ield Kis called a comple e disc e e alua ion ield
i e e y Cauchy sequence in Kcon e ges in K. A disc e e alua ion ield ˆ
K
wi h alua ion ˆνis called a comple ion o Ki i is comple e, ˆν|K=νand K
is a dense sub ield o ˆ
Kwi h espec o dν. This comple ion is unique up o
isomo phism.
P oposi ion 2.5. E e y disc e e alua ion ield Khas a unique comple ion
up o K-isomo phism.
P oo . We shall p o e ha he ield A/min p e ious lemma is he unique
comple ion o K. Fo ha , i s no e ha Kis embedded in A/mby he
na u al map a→(a) + m. Now, o a Cauchy sequence (an)in Kand any
eal numbe M, he e exis s a posi i e in ege n0such ha o m, n ≥n0,
ν(am−an)≥M. Thus, i we ake (an0)which clea ly con e ges in K, we ge
ˆν((an0)−(an)) ≥M, p o ing ha Kis dense in A/m. To p o e comple e-
ness, le ((a(m)
n)n)mbe a Cauchy sequence in A/m(wi h espec o dˆν). Le
n1, n2, . . . be an inc easing sequence o posi i e in ege s such ha o i, j ≥nm,
ˆν(a(m)
i−a(m)
j)≥M. Then, (a(m)
nm)mis a Cauchy sequence in Kand he limi o
((a(m)
n)n)min A/m(wi h espec o dˆν). This p o es A/mis a comple ion o K.
Fo uniqueness, assume (ˆ
K1,ˆν1)and (ˆ
K2,ˆν2) o be wo comple ions o
K. Le 1Kbe he iden i y map in K. Then, we ex end his isomo phism by
con inui y om K, as a dense sub ield o ˆ
K1, o ˆ
K1. This means, o an elemen
a∈ˆ
K1, we ake a Cauchy sequence (an)in Ksuch as lim1an=aand we
map i o b= lim2an∈ˆ
K2. This map is well de ined. Fo ha no e ha i
we conside wo dis inc Cauchy sequences con e ging o a,(an)and (a0
n), by
13
2.3. P ime ideals in ini e ex ensions o global and local ields
Las ly, no e ha K×is gene a ed by he p ime elemen s in K. Fo ha ,
no e ha a p ime elemen is o alua ion 1 and since a uni in he alua ion
ing has alua ion 0, hei p oduc is again a p ime; hus, all elemen s o
alua ion 0 can be ob ained as di ision o p imes and any nonze o elemen
can be ob ained om p imes, as a powe o a p ime imes an elemen o null
alua ion. This will be impo an in Chap e 4 o educe p oo s o he case o
a p ime elemen .
20
Chap e 3
The B aue G oup
In he axioma ic de ini ion o a ield K, we assume K o be a commu a i e
ing. We may elax his de ini ion no asking o commu a i i y. I we do so,
we ge a kind o non (necessa ily) commu a i e ields. These will be called
di ision algeb as o skew ields and hey a e no hing bu iden i y ings whe e
di ision is possible, i.e. all non ze o elemen s a e in e ible. Mos o he di-
ision algeb as seen in unde g adua e cou ses a e usually commu a i e and
hus, usual ields. The i s example o a non commu a i e di ision algeb a
was he so-called qua e nion algeb a, H, disco e ed by William Rowan Hamil-
on (1805-1865) in 1843 while walking along he Royal Canal in Dublin (he
ca ed he de ining o mula o qua e nions i2=j2=k2=ijk =−1in o he
s one o B oome B idge in an impulse a e yea s o hinking). Qua e nions
came o exis ence as an e o o unde s and o a ions in a h ee dimensional
space, jus as complex numbe s desc ibe o a ions in wo dimensions.
The heo y p esen ed in his chap e is u he iche and mo e ex ensi e
han he one gi en in ou p esen a ion. We ha e de eloped jus a minimal
amoun o heo y due o space cons ain s. Thus, he in e es ed eade is
s ongly encou aged o check [Jac85], o example, o mo e de ails.
E en i no s a ed explici ly, all k-algeb as in his wo k a e assumed o be
associa i e.
3.1 Cen al simple algeb as and he B aue g oup
Le kbe a ield and Aak-algeb a. I he cen e o Ais exac ly k,Ais said
o be cen al o e kand i he only ( wo-sided) ideals o Aa e 0 and Ai sel ,
Ais said o be simple. I Ais bo h cen al o e kand simple, Ais called a
cen al simple algeb a o e k. As an example o cen al simple algeb as we
ha e di ision algeb as o e hei cen e . Fo ins ance, His a cen al simple
21
3.1. Cen al simple algeb as and he B aue g oup
algeb a o e R.
Ou aim is o de ine a g oup, o a ield k, whose elemen s will be some
simila i y classes de ined upon simple cen al algeb as o e k. To de ine a
g oup, we need an ope a ion, and his ope a ion will be based on he enso
p oduc . Then, we need i s o de ine he enso p oduc o wo ec o spaces.
Fo ha , le Aand Bbe wo k- ec o spaces. A balanced p oduc o Aand
Bis de ined o be an abelian g oup G oge he wi h a map :A×B→G
sa i ying o all a, a0∈A,b, b0∈Band λ∈k,
1. (a+a0, b) = (a, b) + (a0, b),
2. (a, b +b0) = (a, b) + (a, b0),
3. (λa, b) = (a, λb).
I is deno ed as (G, ). I (G0, 0)is ano he balanced p oduc , a mo -
phism om (G, ) o (G0, 0)is a g oup homomo phism η:G→G0such
ha 0= η. Now, he enso p oduc o Aand Bis a balanced p oduc
(A⊗kB, ⊗)such ha o any o he balanced p oduc (G, ), he e exis s
a unique mo phism om (A⊗kB, ⊗) o (G, ), i.e. (A⊗kB, ⊗k)is uni e -
sal o his p ope y. An explici cons uc ion o he enso p oduc ia he
ca esian p oduc A×Bwhe e i s elemen s a e w i en as sums o he ele-
men a y enso s a⊗kbwi h (a, b)∈A×Bis gi en in P oblem A.6. No e
dimk(A⊗kB) = dimk(A) dimk(B)and ha A⊗kBcan be endowed wi h a
na u al p oduc (a⊗kb)(a0⊗kb0)=(aa0⊗kbb0), making A⊗kBak-algeb a.
Now, we need o de ine he a o emen ioned simila i y ela ion on cen al
simple algeb as o e a ield k. Wi h ha goal in mind, we s a e and p o e
a c i e ion o know unde which condi ions can an algeb a o e a ield kbe
ac o ed as he enso p oduc o wo k-subalgeb as. We conside jus he ini e
dimensional case.
P oposi ion 3.1. Le Aand Bbe subalgeb as o a ini e dimensional algeb a
Do e a ield k. Then, D∼
=A⊗kBi he ollowing condi ions a e sa is ied.
1. ab =ba o all a∈Aand b∈B.
2. D=AB and [D:k]=[A:k][B:k].
P oo . The i s condi ion ensu es he map ϕ:A⊗kB→Dmapping a⊗b→
ab is a ing homomo phism and k-linea . Fo he sake o illus a ion we shall
show ha he p oduc is p ese ed hanks o i s condi ion, o he p ope ies
a e easy o check om he de ini ion o he enso p oduc and a e le o he
eade .
ϕ((a⊗b)(a0⊗b0)) = ϕ(aa0⊗bb0) = aa0bb0=aba0b0=ϕ(a⊗b)ϕ(a0⊗b0).
22
Chap e 3. The B aue G oup
The second condi ion implies ϕis su jec i e; hus, an isomo phism by
dimension coun ing o ec o spaces.
Le us see a di ec applica ion o his c i e ion.
P oposi ion 3.2. Le Abe a k-algeb a. Then, Mn(A)∼
=Mn(k)⊗kA. In
pa icula , Mmn(k)∼
=Mm(k)⊗kMn(k).
P oo . I is s aigh o wa d o check ha he subalgeb as Mn(k)and A1n
whe e 1nis he iden i y in Mn(A), sa is y he condi ions in P oposi ion 3.1.
Jus no e Ma1m=a1mM o any M∈Mn(k)and any a∈Aand ha any
M∈Mn(A)can be exp essed uniquely as an A-linea combina ion o elemen s
o a gi en basis {eij} o Mn(k)since a k-basis o Mn(k)au oma ically gi es
an A-basis o Mn(A). Las asse ion ollows now om he i s one by aking
A:= Mm(k).
Now, we a e in posi ion o de ine he simila i y ela ion. Le Aand B
be wo ini e dimensional cen al simple algeb as o e k. We will say Aand
Ba e simila and w i e A∼Bwhen Mn(A)∼
=Mm(B) o some posi i e
in ege s n, m. This simila i y ela ion is clea ly e lexi e and symme ic. To
see ansi i i y, le Mn(A)∼
=Mm(B)and Ml(B)∼
=M (C), hen
Mnl(A)∼
=Mnl(k)⊗A∼
=Mn(k)⊗Ml(k)⊗A∼
=Ml(k)⊗Mn(A)
∼
=Ml(k)⊗Mm(B)∼
=Ml(k)⊗Mm(k)⊗B∼
=Mm(k)⊗Ml(B)
∼
=Mm(k)⊗M (C)∼
=Mm(k)⊗M (k)⊗C∼
=Mm (k)⊗C
∼
=Mm (C),
whe e we ha e used associa i i y and commu a i i y o he enso p oduc and
he o mulas ob ained in P oposi ion 3.2. Thus, ∼is an equi alence ela ion
and we may conside equi alence classes
[A] = {B ini e dimensional simple cen al algeb a : B∼A}.
Now we shall de ine a bina y ope a ion ia he enso p oduc o he
se o hese equi alence classes. Le A∼A0and B∼B0. We claim now
A⊗B∼A0⊗B0. Since A∼A0and B∼B0we know by de ini ion o ∼ ha
Mn(A)∼
=Mm(A0)and Ml(B)∼
=M (B0), o equi alen ly by P oposi ion 3.2,
A⊗Mn(k)∼
=A0⊗Mm(k)and B⊗Ml(k)∼
=B0⊗M (k) o posi i e in ege s
n, m, l, . Then
Mnl(A⊗B)∼
=A⊗B⊗Mnl(k)∼
=A⊗Mn(k)⊗B⊗Ml(k)
∼
=A0⊗Mm(k)⊗B0⊗M (k)∼
=A0⊗B0⊗Mm (k)
∼
=Mm (A0⊗B0),
p o ing ou claim. This means ha he bina y ope a ion [A] + [B] := [A⊗B]
is well de ined.
23
3.1. Cen al simple algeb as and he B aue g oup
The opposi e algeb a o A, deno ed Aop, is de ined by dualizing he p oduc
in A, i.e. e e sing he p oduc in Ao , in o he wo ds, assigning he elemen
ba o he p oduc a·b. The en eloping algeb a o A, deno ed Ae, is de ined as
he enso p oduc Ae=A⊗kAop.
To make his se o equi alence classes in o an abelian g oup we need
o p o e i s associa i i y, commu a i i y, exis ence o an iden i y and an
in e se o all equi alence classes. Associa i i y and commu a i i y ollows
di ec ly om associa i i y and commu a i i y o he enso p oduc . No e
ha Mn(A)∼
=A⊗Mn(k); hus, A∼A⊗Mn(k). Hence, [Mn(k)] = 0 ac s as
he iden i y o +. We shall see [A]+[Aop]=0and [Aop]ac s as he in e se
o [A]wi h espec o +, o equi alen ly by de ini ion, ha he en eloping
algeb a ac s always as he iden i y.
P imi i e ings and he Densi y Theo em
Fo an abelian g oup M, he se o endomo phisms o M,EndZMo End M,
has a na u al ing s uc u e. Wi h ing o endomo phisms, we mean a sub ing
o End M o an abelian g oup M. We de ine a ing ep esen a ion, as a ing
homomo phism ρ:R→End M o an abelian g oup M. A ep esen a ion ρo
Rac ing on M, i.e. i s image is in End M, yields a le R-module s uc u e o
Mby de ining am =ρ(a)m o any a∈Rand m∈M. Con e sely, gi en a le
R-module M,Mis an abelian g oup and we can de ine ρ=ρM:R→End M
ia he assignmen a→aM o any a∈R, whe e aM∈End Mis le mul i-
plica ion by a. Thus, ρis a ing ep esen a ion. Fo an R-module M,EndRM
will deno e he g oup o R-linea endomo phisms.
An i educible ep esen a ion o a ing is a ep esen a ion such he module
Mis nonze o and whose only submodules a e 0 and i sel . We may say Mis
R-i educible i he e is such a ep esen a ion and i is comple ely educible i
i is he di ec sum o i educible R-modules. The ke nel o a ep esen a ion
ρis called he annihila o o M,AnnR(M) := { ∈R: m = 0,∀m∈M}. I
AnnR(M) = ke ρ= 0, we say ρ(o some imes he R-module M) is ai h ul.
This ke nel is ob iously an ideal o R. A ing Ris called (le ) p imi i e i i
has an i educible and ai h ul ep esen a ion.
The s uc u e o p imi i e ings is o ally de e mined by he impo an
Densi y Theo em om Na han Jacobson ([Jac85], p. 199). Fo ou means we
only need he pa ial esul on ini e dimensional case ha gi en a p imi i e
ing Rac ing on an abelian g oup M,Ris isomo phic o he ini ely dimen-
sional ec o space1End∆Mwhe e ∆ = EndRM. Thus, when e e ing o he
Densi y Theo em we will be e e ing o his pa ial esul . No e i educibili y
1We shall use he e m ec o space o modules o e a di ision ing and no jus o e
ields, jus o ag ee wi h he e minology in [Jac85].
24
Chap e 3. The B aue G oup
o Mis jus needed o ensu e ∆ = EndRMis a di ision algeb a ia Schu ’s
Lemma. Then, i we can ensu e his ing o endomo phisms is a di ision alge-
b a, i is enough Mbeing comple ely educible. Fo a mo e in de ail discussion
on his see [Jac85].
We ha e a na u al module ac ion o Aeon Ade ined by (Pai⊗a0
i)x=
Paixa0
i. Di ec e i ica ion shows i is a well de ined module ac ion. Ae-
submodules o Aa e wo-sided ideals o A; hus, i Ais simple Ais Ae-
i educible.
Rega ding Aas a le ( igh ) A-module in he na u al way, he elemen s o
EndAAa e he igh (le ) mul iplica ion maps x7→ xa (x7→ ax) since i an
endomo phism maps 17→ a, hen, (x) = (x·1) = xa. No e he e e sing
o le and igh . Then, EndAeAis he se o maps ha a e bo h le and igh
mul iplica ions. Jus no e ha i maps 17→ a hen, (x) = (1 ·1·x) =
ax =xa = (x·1·1) = (x). These a e p ecisely he ones x7→ cx whe e cis
in he cen e o A. I Ais cen al o e k, hen x7→ αx, whe e α∈k.
Theo em 3.3. Le Abe a ini e dimensional cen al simple algeb a o e a
ield k. Then, Ae=A⊗kAop ∼
=Mn(k)whe e n= dimkA.
P oo . Rega ding Aas an Ae-module as abo e, Ais Ae-i educible and
EndAeA=k o being simple and cen al o e k espec i ely. Since Ais
ini e dimensional o e k, by he Densi y Theo em Aemaps on o EndkA.
Now, since bo h ec o spaces a e o dimension n2o e k(no e dimk(Ae) =
dimk(A) dimk(Aop) = n2), i is an isomo phism Ae∼
=EndkA∼
=Mn(k).
Ae∼
=Mn(k)is known o be simple (see [G i07], P oposi ion 1.4, p. 360).
Thus i is simple cen al o e kand by Theo em 3.3, [Aop]ac s as he in e se
o [A]wi h espec o +. Thus, i is only le o check his se o equi alen
classes is closed unde he ope a ion +.
Theo em 3.4. Le Abe a ini e dimensional cen al simple subalgeb a o an
algeb a B. Then, B∼
=A⊗kCwhe e Cis he cen alize o Ain B. The
ideals o Ba e in co espondence wi h he ideals o Cby he bijec ion a→Aa.
Mo eo e , he cen e o Bcoincides wi h he cen e o C.
P oo . We shall use P oposi ion 3.1. Since Aeis simple Bis a di ec sum o
i educible Ae-modules and o Abeing Ae-i educible, hey a e all isomo phic
o A(no e ha wo i educible Ae-modules a e always isomo phic since i Mis
an i educible R-module hen M∼
=R/m o a maximal ideal mand since Ris
simple hey a e all isomo phic, see P oblem A.7). Now, no e ha he gene a o
o Aas an Ae-module, 1, sa is ies he condi ion (a⊗k1)1 = a1=1a= 1(1⊗ka)
and (a⊗k1)1 = 0 implies a= 0. Thus, since all i educible Ae-modules a e
isomo phic we may choose an elemen cjin each i educible Ae-module sa is-
ying (a⊗k1)cj=acj=cja=cj(a⊗k1) and (a⊗k1)cj= 0 impliyng a= 0.
25
3.1. Cen al simple algeb as and he B aue g oup
Applying his o Bas an Ae-module and no ing he map om A o each i -
educible Ae-module mapping 17→ cjex ends o an isomo phism by linea i y,
we may w i e B=LAcjwhe e acj=cja o all a∈Aand acj= 0 implies
a= 0. Then, clea ly cj∈Cand any elemen o Bcan be uniquely w i en
as a ini e sum Pajcj o aj∈A. Fo any c∈C,c=Pajcj, bu ac =ca
implies aaj=aja o any a∈A. Thus, aj∈k( o Abeing cen al o e k)
and c∈Pkcj. Hence, C=Pkcjand cjis a basis o Cand clea ly B=AC
and [B:k]=[A:k][C:k]. Thus, by P oposi ion 3.1, B∼
=A⊗kC, as wan ed.
Now, le abe an ideal in C. Then, Aais an ideal in B=AC. We claim ha
Aa∩C=a. Le βA={x1= 1, . . . , xn}be a k-basis o A. Since B∼
=A⊗kC,
any elemen in Bcan be uniquely w i en as a C-linea combina ion o he
k-basis βA. Thus, he elemen s o Aaa e a-linea combina ions o βA. In pa -
icula , elemen s bo h in Aaand in Ca e o he o m c1x1=c1∈a. Hence,
Aa∩C=aas claimed. This p o es ha he map a7→ Aais injec i e since
Aa=Abimplies a=bby aking he in e sec ion wi h C. To check su jec i -
i y, le bbe an ideal o B. Then, bis an Ae-submodule o B. Hence, b=PAbj
whe e bj∈a:= b∩C. This implies, b=Aa, p o ing su jec i i y. Thus, we
ge a one- o-one co espondence be ween he ideals o Cand hose o B.
Las ly, we show he cen e o Bcoincides wi h he cen e o C. Clea ly he
cen e o Bis con ained in Cand hus, in he cen e o C. Fo he con e se,
any elemen in he cen e o Ccommu es wi h e e y elemen o B=AC and
hus, i is in he cen e o B.
Thus, when Cis simple and cen al o e k,Bis simple and cen al o e k.
Co olla y 3.5. The enso p oduc o wo ini e dimensional cen al simple
algeb as o e a ield kis again a ini e dimensional cen al simple algeb a
o e k. In gene al, he enso p oduc o a ini e numbe o ini e dimensional
cen al simple algeb as o e a ield kis again a ini e dimensional cen al
simple algeb a o e k.
Then, he se o equi alen classes is closed unde +and i can be ega ded
as an abelian g oup. This g oup is called he B aue g oup o kand i is de-
no ed by B (k). The B aue g oup was i s in oduced by Richa d B aue
(1901-1977) in 1929.
Now, le Lbe a ield ex ension o k. Then, o any k-algeb a A,A⊗kLcan
be ega ded as an L-algeb a. This L-algeb a is deno ed as ALand is called
he algeb a ob ained om Aby ex ending he base ield o L. I Ais ini e
dimensional cen al simple o e k, i can be seen as a co olla y o Theo em
3.4 ha ALis ini e dimensional cen al simple o e L. Le Abe a ini e di-
mensional cen al simple algeb a o e k. A ield Lis called a spli ing ield o
26
Chap e 3. The B aue G oup
Ai AL=A⊗kL∼
=Mn(L) o some posi i e in ege n.
Now, le Lbe a ini e ex ension o k. Then, i Ais ini e dimensional cen-
al simple o e k,ALis ini e dimensional cen al simple o e L. We ha e
(A⊗kB)L∼
=AL⊗LBLand Mn(k)L∼
=Mn(L). Thus, a g oup homomo phism
may be de ined om B (k) o B (L)mapping [A]7→ [AL]. The ke nel o his
homomo phism, i.e. he classes [A]in B (k)such ha AL∼0, a e exac ly he
classes o k-algeb as wi h spli ing ield L. This ke nel o ms a subg oup ha
is deno ed by B (L/k).
Recall ha o a ield Fand an n-dimensional F- ec o space V,EndF(V)∼
=
Mn(F). I commu a i i y is no assumed, we should ake he opposi e ing
when passing om endomo phisms o ma ices (see P oblem A.8).
Lemma 3.6. Le Abe a cen al simple algeb a o e kand L/k a ini e ield
ex ension such ha Lis a sub ield o Mn(A)∼
=EndAop V o Van Aop- ec o
space. Then, he cen alize o Lin Mn(A)coincides wi h EndAop⊗kLV.
P oo . We shall see he cen alizing Lcondi ion is equi alen o Aop ⊗kL-
linea i y ega ding Vas an Aop ⊗kL-module ia he ac ion (d⊗kl)x=dlx =
ldx. Le l∈L⊆EndAop V. Then, lc =cl o some endomo phism cmeans
lc( ) = c(l( )) o all ∈V. Bu cis Aop-linea ; hus, c((l⊗ka) ) = c(la ) =
lac( )=(l⊗ka)c( )and i is Aop ⊗kL-linea . The same easoning p o es he
con e se.
Theo em 3.7. Le ∆be a ini e dimensional cen al di ision algeb a o e k.
Then, a ini e ex ension L/k is a spli ing ield o ∆i and only i Lis a
sub ield o an algeb a A=Mn(∆) such ha CA(L) = L.
P oo . We jus need he only i pa , o a p oo o he con e se see [Jac85].
Le Lbe a sub ield o A=Mn(∆) which is sel -cen alised. Recall Amay be
iden i ied wi h End∆0V o an n-dimensional ec o space Vo e ∆0= ∆op.
Then, we ega d Vas an ∆0⊗kL-module as be o e. Since bo h ∆0and L
a e simple, by Co olla y 3.5, ∆0⊗kLis simple oo and his ac ion is nec-
essa ily ai h ul. Now, since ∆0⊗kLis ini e dimensional o e k,Vis com-
ple ely educible as a ∆0⊗L-module. By p e ious lemma, he cen alize
o Lin Ais End∆0⊗LVand his is Lby assump ion, which is a ield;
hus, a di ision algeb a. Then, we may apply he Densi y Theo em o ob-
ain ∆0⊗L∼
=EndLV∼
=M (L)and Lis a spli ing ield o ∆0; hence, o
∆.
27
3.2. G oup cohomology, c ossed p oduc s and cyclic algeb as
3.2 G oup cohomology, c ossed p oduc s and
cyclic algeb as
Gi en a g oup Gand an abelian g oup M, we say Mis a (le ) G-module i
we can de ine a (le ) G-ac ion on M, i.e. a map G×M→M, such ha o
any g, h ∈Gand any n, m ∈Mwe ha e
g(m+n) = gm +gn, g(hm)=(gh)m, 1m=m.
Now, le Gbe a g oup and MaG-module. We de ine he g oup o n-
cochains, deno ed as Cn(G, M)as he se o maps om Gn o M o any
n∈N. Fo n= 0,C0(G, M) = Mby con en ion. This se can be endowed
wi h a g oup ope a ion by de ining he sum o maps componen wise. This way,
i becomes an abelian g oup.
Le us de ine he maps δn:Cn(G, M)→Cn+1(G, M) ia he assignmen s,
δn( )(σ1, ..., σn+1) =σ1 (σ2, ..., σn+1)
+
n
X
j=1
(−1)j (σ1, ..., σjσj+1, ..., σn+1)
+ (−1)n+1 (σ1, ..., σn),
o n∈Nand δ0(m)(σ) = σm −m o n= 0. The maps δnde ine g oup
homomo phisms and δ2=δnδn+1 is he i ial map. Comp oba ion o hese
ac s is s aigh o wa d bu edious; hus, i is le o he eade . The maps δn
a e called di e en ials.
Since δ2= 0,Im δn−1is inside ke δn, and i makes sense o de ine he
quo ien g oup Hn(G, M) = Zn(G, M)/Bn(G, M), whe e Zn(G, M) := ke δn
and i s elemen s a e called n-cocycles and Bn(G, M) := Im δn−1and i s el-
emen s a e called n-cobounda ies. Fo n= 0 we de ine B0(G, M)=0. The
g oup Hn(G, M)is called he n h cohomology g oup o Gwi h coe icien s in
M. Two n-cocycles a e called cohomologous i hey a e equal up o a cobound-
a y, i.e. i hey ep esen he same elemen o Hn(G, M).
The G-in a ian pa o Mis de ined as he subse o M ha is in a ian
unde he ac ion o G, i.e. MG={m∈M:σm =m, σ ∈G}. Whene e
Gis cyclic wi h gene a o σ, he no m g oup,N(G), is de ined as he se
N(G) := {Pσkm:m∈M}. In his cyclic case, he second cohomology g oup
can be desc ibed by hese cons uc ions (P oblem 16 in [Mo 96], p. 105-106).
Theo em 3.8. Le Gbe a cyclic g oup and MaG-module. Then, H2(G, M)∼
=
MG/N(G).
P oo . See P oblem A.9.
28
Chap e 3. The B aue G oup
We in oduce now a way o cons uc ing a k-algeb a based on a ini e
Galois ield ex ension. Le L/k be a ini e Galois ield ex ension and le {xσ}
be a collec ion o symbols in 1-1 co espondence wi h he elemen s o G:=
Gal(L/k). We ega d he k-algeb a Aas a ec o space o e Lwi h basis {xσ},
i.e. A=LLxσ, and we de ine a p oduc on Aby he ela ions
xσxτ=κσ,τ xστ , xσl=σxσ,∀l∈L,
whe e κσ,τ a e elemen s o L×and Lis ega ded as a G-module by he na u al
ac ion o G, i.e. σl =σ(l) o any σ∈Gand any l∈L.
Since {xσ}is an L-basis o A, any elemen o Acan be uniquely ep-
esen ed as a ini e sum Plσxσwi h coe icien s in L. Thus, he p oduc o
wo elemen s Plσxσand Pl0
σxσis de ined as Pκσ,τ lσσl0
τxστ , by he abo e
ela ions.
To make Ain o an associa i e k-algeb a, he κs, mus be chosen app o-
p ia ely. Since
(xσxτ)xρ=κσ,τ xστ xρ=κσ,τ κστ,ρxσστρ
xσ(xτxρ) = xσκτ,ρxτρ = (σκτ,ρ)κσ,τρxστρ,
and o ensu e associa i i y we need, κσ,τ κστ,ρ = (σκτ,ρ)κσ,τρ, which is exac ly
he 2-cocycle condi ion we ha e seen be o e in mul iplica i e no a ion o he
map κ:G×G→L×,(σ, τ)7→ κσ,τ . Dis ibu i e laws a e s aigh o wa d
o check om de ini ion. Since by he 2-cocycle condi ion κ1,σ =κ1,1and
κσ,1=σκ1,1, he elemen 1 = κ−1
1,1x1is he iden i y o he mul iplica ion as
a simple compu a ion shows. Finally, since he elemen s o ka e ixed by all
k-au omo phisms o L, he ing mul iplica ion and he k-scala mul iplica ion
as a ec o ial space a e compa ible, i.e. λ(ab) = (λa)b=a(λb) o all λ∈k
and a, b ∈A. Thus, Ais a k-algeb a. We shall deno e Ain he ollowing as
(L, G, κ). We shall see hese k-algeb as a e simple cen al o e kand iden i y
he subg oup B (L/k)wi h he second cohomology g oup H2(G, L×).
P oposi ion 3.9. Le A= (L, G, κ). Then, Ais a simple cen al algeb a o e
kand [A:k] = n2whe e n= [L:k]. Rega ding Las a sub ield o A(L1A)i
coincides wi h i s cen alize in A, i.e. CA(L) = L.
P oo . The ela ion [A:k] = [A:L][L:k] = [L:k]2=n2holds om
he de ini ion o he c ossed p oduc . Now we shall see Ais simple. Fo ha ,
we shall see i s all xσa e in e ible. The p oduc xσxσ−1=κσ,σ−1κ1,11is
a nonze o elemen o L; hus, xσis in e ible in A. Now, le abe a p ope
ideal o A. We shall see i is i ial. We w i e a=a+a o a∈A. Then,
A=A/a6= 0 since i is p ope . Hence, he usual p ojec ion es ic ed o L,
l7→ lis a monomo phism in o A, since Lis a ield and his homomo phism
29
4.2. Local ields
o he p-adic opology in K×a e p ecisely hose no m g oups, and i s p o ini e
comple ion being isomo phic o he Galois g oup Gal(Kab/K).
4.2 Local ields
Theo em 4.1 (Local Class Field Theo y). Le Kbe a local ield. Then,
1. The e exis s a unique con inuous homomo phism
ρK:K×→Gal(Kab/K),
sa is ying he ollowing condi ions.
a) Fo a ini e abelian ex ension Lo K,ρKinduces an isomo phism
K×/NL/K(L×)∼
=
→Gal(L/K).
b) I Kis a comple e disc e e alua ion ield wi h ini e esidue ield
Fq, hen he diag am
K×Gal(Kab/K)
ZGal(Fab
q/Fq).
ρK
νK
ρFq
is commu a i e, whe e νKis he disc e e alua ion in Kand he
map Gal(Kab/K)→Gal(Fab
q/Fq)is he composi ion
Gal(Kab/K)→Gal(Ku /K)∼
=
→Gal(Fab
q/Fq),
whe e Gal(Kab/K)→Gal(Ku /K)is he es ic ion o au omo -
phism o Kab o Ku .
2. The e is a one- o-one co espondence h ough ρKbe ween open subg oups
o Gal(Kab/K)and open subg oups o ini e index o K×, i.e. ini e
abelian ex ensions o Klie in one- o-one co espondence wi h open sub-
g oups o ini e index o K×.
The emainde o he sec ion is de o ed o p o ing his impo an heo-
em. We will assume a some poin s Khas cha ac e is ic 0 o he sake o
simplici y, bu he esul s a e s ill alid in posi i e cha ac e is ic e en i he
p oo s end o be mo e edious. The cases K=Rand Ca e deal sepa a ely
(see P oblem A.12). Now, le Kbe a comple e disc e e alua ion ield wi h
ini e esidue ield.
36
Chap e 4. Class Field Theo y
P oposi ion 4.2. Le Kbe a local ield and La ini e sepa able ex ension o
K. Then,
1. The ollowing diag am is commu a i e.
B (K)Q/Z
B (L)Q/Z.
in K
mul iplica ion by [L:K]
in L
2. The o de o B (L/K)is exac ly [L:K].
P oo . Fi s no e second asse ion ollows om he i s one. Since in is an
isomo phism, he ke nel o he mul iplica ion by [L:K]in Q/Zis isomo phic
o he ke nel o he es ic ion B (K)→B (L), which we deno ed by B (L/K).
Since he i s ke nel is p ecisely {n/[L:K] + Z}and has o de [L:K],
B (L/K)has o de [L:K] oo. Now we p o e he i s asse ion. Le eand
be he ami ica ion index and esidue deg ee o Lo e K espec i ely. By
P oposi ion 2.9 we ha e [L:K] = e . Le πbe a p ime elemen in Land πe
a co esponding p ime elemen in Kand ecall om Lemma 1.7 he ollowing
diag am is commu a i e
X(Fq)Q/Z
X(Fq )Q/Z.
es ic ion mul iplica ion by
whe e he ho izon al a ows a e mapping χ7→ χ(σ)and χL7→ χ(σ ), whe e
χLis he es ic ion o χ o X(Fq ) iewed as an elemen o X(L)and σ
and σ a e he F obenius au omo phisms in he co esponding absolu e Galois
g oups. Now he i s asse ion ollows om his since he isomo phism in Kis
mapping (χ, πe)7→ χ(σ)whils in Lis mapping (χ, π)7→ χ(σ ). Then, since
[L:K] = e , we see (χ, πe)7→ χ(σ)7→ e χ(σ)and (χ, πe)7→ (χL, πe) =
(χL, πe) = e(χL, π)7→ eχ(σ ) = e χ(σ)coincide p o ing he commu a i i y
o he diag am in he i s asse ion.
P oposi ion 4.3. Fo each ini e abelian ex ension Lo K, he e is an iso-
mo phism
K×/NL/K(L×)∼
=
→Gal(L/K)∗∗ ∼
=
→Gal(L/K),
gi en by he composi ion α7→ (χ→in K(χ, α)) 7→ σ.
P oo . Fi s no e he map K×→Gal(L/K)∗∗ →Gal(L/K)gi en by he
composi ion α7→ (χ7→ in K(χ, α)) 7→ σ, has ke nel con aining he no m
37
4.2. Local ields
map. Fo ha , since he las homomo phism is he e alua ion isomo phism
om Pon jagin’s duali y and since in ini e abelian g oups Tχ∈G∗ke χ= 1,
i is enough o check ha he cyclic algeb as (χ, NL/K(L×)) a e i ial in
B (K) o any cha ac e χ∈Gal(L/K)∗. Le Kχ he cyclic ex ension co e-
sponding o he cha ac e χ. Then, by Theo em 3.12, (χ, NKχ/K(K×
χ)) = 0
in B (K). Bu , by he ansi i e p ope y o he no m, we ge NL/K(L×) =
NKχ/K(NL/Kχ(L×)) ⊆NKχ/K(K×
χ)and he induced map K×/NL/K(L×)→
Gal(L/K)is well de ined.
Fo injec i i y i is enough o p o e he inequali y |K×/NL/K (L×)| ≤
|Gal(L/K)|once we p o e i is on o. No e o wo ield ex ensions K⊆E⊆
L,NL/K =NL/ENE/K. Then, E×/NL/E(L×)NE/K
→K×/NL/K(L×)is well
de ined and he sequence
E×/NL/E(L×)NE/K
→K×/NL/K(L×)→K×/NE/K(E×)→1
is exac . Then, |K×/NL/K(L×)|≤|E×/NL/E(L×)||K×/NE/K (E×)|, showing
i is enough o conside ini e ex ensions o p ime deg ee by induc ion on he
p ime ac o s o [L:K]. Bu , in his case he ex ension is cyclic and combining
Theo em 3.12 and p e ious p oposi ion |K×/NL/K(L×)|=|B (L/K)|= [L:
K] = |Gal(L/K)|, which implies |K×/NL/K(L×)|≤|Gal(L/K)| o a sepa a-
ble ex ension L/K. Hence, we a e only le o p o e i is on o o ob ain an
isomo phism. Fo ha , we ha e seen in Sec ion 1.4 ha his homomo phism
will be on o i he only cha ac e annihila ing i s image is he i ial cha ac e .
Such a cha ac e mus sa is y (χ, K×)=0in B (K)by de ini ion. Then, by
Theo em 3.12 we ha e B (Kχ/K)=0and by he second asse ion in P opo-
si ion 4.2, we ha e he o mula [Kχ:K] = |B (Kχ/K)|= 1; hus, Kχ=K
and χ= 0 p o ing su jec i i y.
Now, no e ha o any ini e Galois ex ensions M⊆Lo K, he diag am
K×/NL/K(L×) Gal(L/K)
K×/NM/K(M×) Gal(M/K)
es ic ion
commu es whe e he ho izon al a ows a e p ecisely he ones de ined in he
p e ious p oposi ion and he le e ical a ow is a usual p ojec ion since
NL/K(L×)⊆NM/K(M×). Then, he isomo phisms om he p e ious P opo-
si ion a e compa ible wi h he connec ion homomo phisms and by he uni e -
sal p ope y o he in e se limi hey induce a homomo phism ρK:K×→
Gal(Kab/K),α7→ σ:= (σL)L. This is he celeb a ed homomo phism o local
class ield heo y. We shall show i has he p ope ies desc ibed in Theo em
4.1.
38
Chap e 4. Class Field Theo y
P oposi ion 4.4. Le Kbe a comple e disc e e alua ion ield wi h ini e
esidue ield. Then, ρKhas he p ope y (b) in Theo em 4.1(1).
P oo . Le us see commu a i i y o he diag am o a p ime elemen πsince
K×is gene a ed by p ime elemen s. The alua ion o a p ime elemen is 1
and 1 gene a es Zas a g oup and Gal(Fab
q/Fq)is isomo phic o he p ocyclic
p o ini e comple ion o Z,b
Z, gene a ed opologically by he F obenius au o-
mo phism x7→ xq. Then, he image o he p ime π h ough he composi e
νKρFqis he F obenius au omo phism in Gal(Fab
q/Fq). On he o he way, he
image o π h ough ρKsa is ies χ(ρK(π)) = in K(χ, π)=1/n +Z o each
un ami ied cha ac e χ, whe e nis he index o he ke nel o χin Gal(Kab/K).
This elemen o he bidual is mapped ia he e alua ion isomo phism o an
au omo phism σin Gal(Kab/K)such ha χ(σ) = 1/n +Z o each un am-
i ied cha ac e , i.e. i induces gene a o s in each cyclic un ami ied ex ension
Kχ; hus, i s es ic ion is a gene a o o Gal(Ku /K), which is mapped o
he F obenius au omo phism in Gal(Fab
q/Fq)by he canonical isomo phism
Gal(Ku /K)∼
=Gal(Fab
q/Fq), p o ing commu a i i y o he diag am.
P oposi ion 4.5. Fo a local ield K,ρKis con inuous.
P oo . Assume o simplici y cha K= 0. Since Gal(Kab/K)is a p o ini e
g oup, o check con inui y o he map ρK:K×→Gal(Kab/K)i is enough
o show con inui y o each induced map K×→Gal(L/K), by he uni e sal
p ope y o he in e se limi no ing we a e wo king o e he ca ego y o opo-
logical g oups. Since each ini e Galois g oup is disc e e, i is enough o check
he ke nel is open by Lemma 1.2. Since he ke nel is o ini e index o he
image g oup being ini e, i ollows om P oposi ion 2.18 i is open.
Rema k. We assumed K o be o null cha ac e is ic in o de o apply P opo-
si ion 2.18.
Now, we show he map ρKis unique in he sense o Theo em 4.1.
P oposi ion 4.6. Le ˜ρ:K×→Gal(Kab/K)be an homomo phism sa is y-
ing,
1. Le Lbe a cyclic ex ension o K. Then, he composi e map,
Kטρ
→Gal(Kab/K)→Gal(L/K),
maps NL/K(L×) o {1}.
2. Le Lbe a ini e un ami ied ex ension o K. Then, he image o a p ime
by he same composi e map o (1), is a gene a o o he cyclic Galois
g oup.
Then, ˜ρ=ρK.
39
4.2. Local ields
P oo . As be o e, i is enough o p o e ρ0(π) = ρK(π) o p ime elemen s
π∈K×, as hey gene a e he g oup o uni s. We shall see χ(ρ0(π)) = χ(ρK(π))
o all χ, which is easie o check and implies p e ious equali y since he iso-
mo phism in Pon jagin’s duali y is he e alua ion isomo phism. Le nbe he
o de o χ(ρK(π)). Le Kn/K be he unique un ami ied ex ension o deg ee
n. Then, ρK(π) es ic s o a gene a o o Gal(Kn/K). Hence, he e is some
un ami ied cha ac e ψ∈X(K)such ha ψ(ρK(π)) = χ(ρK(π)). Now, le
L/K be he cyclic ex ension co esponding o he cha ac e ψ−χ. Then, he
composi e K×ρK
→Gal(Kab/K)→Gal(L/K)maps π o 1. Thus, by P oposi-
ion 4.4, πis in he no m g oup NL/K(L×). By he i s p ope y o ρ0,ρ0(π)
maps o 1 oo, and we ob ain (ψ−χ)(ρ0(π)) = 0. By he second p ope y,
ψ(ρ0(π)) = ψ(ρK(π)) and we ha e he equali y
χ(ρ0(π)) = ψ(ρ0(π)) = ψ(ρK(π)) = χ(ρK(π)),
concluding he p oo .
Now, o conclude we need o p o e he e is a one- o-one co espondence
be ween abelian ex ensions and open subg oups o ini e index o he g oup
o uni s o he base ield. We ha e seen his is equi alen o ha ing a one- o-
one co espondence be ween open subg oups o ini e index in Gal(Kab/K)
and open subg oups o ini e index in K×. Fo a p o ini e abelian g oup G,
we claim i s open subg oups o ini e index a e in one- o-one co espondence
wi h he ini e subg oups o i s cha ac e g oup G∗gi en ia he assign-
men s H≤G7→ ϕ(H) := {χ∈G∗:χH= 0}and H≤G∗7→ ψ(H) :=
Tχ∈Hke χ(see P oblem A.13). Thus, i we se X(K×) := homcon (K×,Q/Z),
i is su icien o p o e he e is an isomo phism X(K)∼
=X(K×)gi en by
he assignmen χ7→ ρKχ. To p o e injec i i y, jus no e ha he compos-
i e K×ρK
→Gal(Kab/K)→Gal(L/K)is su jec i e o all abelian ex ensions
L/K; hus, since aking he dual homcon (−,Q/Z)is a con a a ian unc-
o mapping an exac sequence K×→Gal(L/K)→0 o an exac sequence
0→Gal(L/K)∗→X(K×), his inclusion is injec i e o he duals o each
ini e Galois g oups; hence, o he dual o he Galois g oup Gal(Kab/K) oo.
Now, we p o e su jec i i y.
Fi s , we shall check Lab is an ex ension o Kab o any sepa able ex ension
L/K. By de ini ion, i is enough o see Kab/L is an abelian ex ension. No e
ha Gal(Kab/K)is abelian and Gal(Kab/L)is one o i s subg oups; hus i is
abelian oo. This shows he es ic ion Gal(Lab/L)→Gal(Kab/K)gi en by
he usual es ic ion o au omo phisms is well de ined. Then, we may de ine
a na u al map X(K)→X(L)by plugging he Galois g oup o he maximal
abelian ex ension o L h ough he usual es ic ion in he le hand side, i.e.
χ7→ χL:= πL/Kχwhe e πL/K deno es his es ic ion.
40
Chap e 4. Class Field Theo y
P oposi ion 4.7. Le Kbe a local ield and La ini e sepa able ex ension o
K. Then, he diag am
L×Gal(Lab/L)
K×Gal(Kab/K).
ρL
NL/K
ρK
is commu a i e, whe e he igh e ical map is he homomo phism ob ained
by es ic ion o au omo phisms o Lab o Kab.
P oo . Le Kbe a comple e alua ion ield wi h ini e esidue ield, since
he eal and complex cases a e easy o check and hus le o he eade .
No e ha , as be o e, i is su icien o p o e in K(χ, NL/K(π)) = in L(χL, π)
o all χ∈X(K), whe e χLdeno es he image o χby he inclusion map
X(K)→X(L). Le be he esidue deg ee o he ex ension L/K. Then, we
may choose an un ami ied elemen ψ∈X(Fq )⊆X(L)such ha (ψ, π) =
(χL, π). No e such an elemen exis s since all he classes o he B aue g oup
con ain such an elemen . Now, since he mul iplica ion by map Q/Z∼
=
X(Fq)→X(Fq )∼
=Q/Zis su jec i e we may choose an un ami ied elemen
ϕ∈X(Fq)⊆X(K)such ha ϕL=ψ. Le L0/L be he cyclic ex ension
co esponding o ϕL−χL. Since (ϕL−χL, π) = 0,πis a no m elemen by
he main p ope y o cyclic algeb as, i.e. π=NL0/L(b) o some b∈(L0)×.
Now, conside he cyclic ex ension K0/K co esponding o he cha ac e ϕ−χ.
Since (ϕ−χ)L0= 0,K0⊆L0. Thus, by ansi i i y o he no m,
NL/K(π) = NL/K(NL0/L(b)) = NL0/K(b) = NK0/K(NL0/K0(b)) ∈NK0/K((K0)×).
Thus, (ϕ−χ, NL/K (π)) = 0 and we ge
in K(χ, NL/K(π)) = in K(ϕ, NL/K (π))
=νK(NL/K(π))(image o ϕby X(Fq)∼
=
→Q/Z)
= ·(image o ϕby X(Fq)∼
=
→Q/Z)
= (image o ϕLby X(Fq )∼
=
→Q/Z)
= in L(ϕL, π) = in L(χL, π),
whe e νKis he disc e e alua ion o Kand second and i h equali ies ollow
om P oposi ion 4.4, hi d om P oposi ion 2.17 and πbeing a p ime elemen
in Land ou h equali y om he map X(Fq)→X(Fq )being mul iplica ion
by . This concludes he p oo .
Lemma 4.8. Le Kbe a local ield o cha ac e is ic 0. Then,
1. Fo n∈N, le Xn(K) := {χ∈X(K) : nχ = 0}and Xn(K×) := {χ∈
X(K×) : nχ = 0}. I Kcon ains a p imi i e n h oo o uni y, we ha e
an isomo phism Xn(K)∼
=
→Xn(K×), gi en by χ7→ ρKχ.
41
4.2. Local ields
2. Le Lbe a ini e ex ension o Kand χ∈X(K×). I NL/Kχ∈X(L×)lies
in he image o X(L)→X(L×),χlies in he image o X(K)→X(K×).
P oo . Le us p o e i s he i s asse ion. We shall see he sequence o
maps
K×/(K×)n→Xn(K)→Xn(K×),
is a sequence o injec i e maps and ha K×/(K×)nand Xn(K×)a e ini e
and ha e same o de , implying he desi ed isomo phism Xn(K)∼
=Xn(K×).
No e he second map is gi en by χ7→ ρKχ, i.e. i is ob ained by plugging he
g oup K×in he le hand side.
Le Kcon ain a p imi i e n h oo o uni y ζn. Then, K(n
√a)is an abelian
ex ension o any a∈K×. Thus, we may de ine a g oup homomo phism
K×→Xn(K) ia he assignmen a7→ χawhe e χa(σ) = /n and is chosen
such ha σ(n
√a) = ζ
nn
√a. This is a well-de ined g oup homomo phism and
no e ha χa= 0 o all a∈(K×)n; hus, i induces a g oup homomo phism
K×/(K×)n→Xn(K). We shall see i is injec i e, i is ac ually an isomo hism
by Kumme Theo y bu we jus need injec i i y o ou means. Fo ha , we
see he ke nel is i ial. Fo a∈K×and χa= 0, we ha e n
√ais ixed by he
Galois g oup, i.e. i is in K×; hence, a∈(K×)nand he ke nel is i ial p o ing
injec i i y. We show in P oposi ion 2.18 ha [K×: (K×)n]is ini e; hus, he
quo ien g oup is ini e. Since Xn(K×)can be iden i ied wi h X(K×/(K×)n)
ia χ(k)↔χ(k(K×)n)and he cha ac e g oup o a ini e abelian g oup has
same o de as he g oup i sel since hey a e isomo phic, al hough no na u-
ally, we ob ain he desi ed isomo phism.
Now we p o e he second asse ion o he lemma. By ansi i i y o he
no m, i is su icien o conside in e media e ields o he ini e abelian ex en-
sion L/K, i.e. we may assume wi hou loss o gene ali y ha L/K is cyclic.
Le G:= Gal(L/K)and conside he ac ion o Go e he g oups X(L)and
X(L×)de ined by σχ : Gal(Lab/L)→Q/Z,τ7→ χ(˜σ−1τ˜σ), whe e ˜σis an
elemen o Gal(Lab/K)whose image in Gal(L/K)is σ o χ∈X(L); and
σχ =σ−1χ o χ∈X(L×) espec i ely o each σ∈G. Now, le χ1∈X(K×)
and assume NL/Kχ1∈X(L×)is he image o χ2∈X(L). Recall om p e i-
ous chap e ha we w i e MG o he elemen s o a G-module Min a ian
unde he G-ac ion. Then, clea ly NL/Kχ1∈X(L×)G, since Galois conju-
ga es ha e he same no m. No e he map X(L)→X(L×)is a homomo -
phism o G-modules, his is easy and le o he eade , and injec i e; hus,
neccesa ily χ2∈X(L)Gno ing ha σNL/Kχ1=NL/Kχ1and using injec-
i i y o ob ain σχ2=χ2. Now le us p o e X(L)Gis con ained in he im-
age o he map X(K)→X(L),χ7→ χL. Le σbe a gene a o o Gand
ix an elemen ˜σ. Then, i pis he o de o Gany elemen o he g oup
Gal(Lab/K)can be uniquely w i en as h˜σj o some h∈Gal(Lab/L)and
42
Chap e 4. Class Field Theo y
some j= 0,1, . . . , p −1since Gal(Lab/K)∼
=Gal(Lab/L)×Gal(L/K). Now,
o χ∈X(L)G, choose an elemen s∈Q/Zsuch ha χ(σ) = ps and de ine he
map χ0:= Gal(Lab/K)→Q/Z ia he assignmen τ=h˜σj7→ χ(h) + js. This
map is clea ly a g oup homomo phism (no e Gal(Lab/K)is abelian and hus
i is easily e i ied χ0(τρ) = χ0(τ)χ0(ρ)) and i induces a map Gal(Kab/K)→
Q/Zand i can be ega ded as an elemen o X(K)and i s image χ0
Lcoincides
wi h χ. Fo ha , no e ha any au omo phism τ∈Gal(Lab/L)has j= 0 in
he abo e o m; hus, χ0
L(τ) = χ(τ)and χ0
L=χ. Thus, he homomo phism
X(K)→X(L)Gis su jec i e and he e exis s an elemen χ3∈X(K)such
ha (χ3)L=χ2. F om p e ious p oposi ion ρKχ3and χ1map o he same
elemen in X(L×) ia NL/K ; hus, χ1−ρKχ3annihila es NL/K(L×). Hence,
he composi ion χ4: Gal(L/K)→Q/Zo χ1−ρKχ3:K×/NL/K(L×)→Q/Z
and he induced isomo phism K×/NL/K(L×)∼
=Gal(L/K)can be seen as an
elemen o X(K)and we ha e χ1=ρK(χ3+χ4), i.e. χ1lies in he image o
X(K)→X(K×)concluding he p oo .
Now, assuming cha K= 0 le χ∈X(K×). We shall see i lies in he image
o X(K)→X(K×). No e bo h g oups a e o sion; hus, le nbe he o de o
χ. We shall assume Kcon ains an n h p imi i e oo o uni y, since K(ζn)/K
is a ini e abelian ex ension, and by Lemma 4.8.2 all cases a e educed o his
one. Then, by Lemma 4.8.1, he map Xn(K)→Xn(K×),χ07→ ρKχ0is an iso-
mo phism o all na u al nand su jec i i y ollows and we a e done wi h he
p oo o he local class ield heo y. As usual, hese isomo phisms a e compa i-
ble wi h usual inclusions whene e ndi ides mand we ob ain an isomo phism
o he di ec limi s, i.e. X(K)∼
=X(K×), concluding he p oo o Theo em 4.1.
To conclude, I would like o highligh again he as onishing beau y o local
class ield heo y: how local ields encode he da a o all hei ini e abelian
ex ensions in hei inne a i hme ic in a a he unexpec ed bu simple way.
This is a nice ep esen a i e o he cha m o algeb a and numbe heo y:
objec s may seem o be so dis an om each o he bu happen o be linked
in an ou o he blue bu easy way. And, gene a ion a e gene a ion mo e
o hese links a e de eloped and we ealize ha e en i we hough we ully
unde s ood a heo y, we we e jus sc a ching he su ace o a whole new wo ld
making you o keep lea ning cons an ly, which, o me, is he mos cap i a ing
aspec o he queen o ma hema ics.
43
Appendix A
Sol ed P oblems
A.1 P elimina ies
P oblem A.1. Show he K ull opology and he p o ini e opology need no
coincide.
Solu ion. We shall ollow he p ocedu e in [Mil20]. I is enough o show
he e is some Galois g oup wi h a leas one subg oup o ini e index non-open.
Le Gal(Q/Q)and he in e media e ield E:= Q(√−1,√2,√3,...,√p, . . . ).
Then, i is an easy exe cise o check G:= Gal(E/K) = lim
←−Gal(Q(√−1,√2,
. . . , √p)/Q)and since each ini e Galois g oup is a ini e p oduc o g oups
Z/2Z,Gis a closed subg oup o he di ec p oduc o a coun able numbe o
g oups Z/2Z. Now, conside he subg oup No Go uples wi h only a ini e
numbe o non- i ial componen s, i.e. a di ec sum o a coun able numbe
o g oups Z/2Z. Also, i is clea ly dense in Gand we may make he quo ien
Γ := G/N, which is a ec o space o e F2. Then, by Zo n’s Lemma Γcon ains
a maximal se o linea ly independen ec o s, which is necessa ily a basis.
Then, ake nelemen s ou o he basis and de ine he subspace spanned by
he emaining se as Gn. Then, Γ/Gnis o dimension no e F2, i.e. o index
2nin Γ. I Gnwe e open in Γ, i would be closed oo, bu ha i is impossible
since Nis dense in G. Then, Gnis o ini e index and non-open, p o ing ou
claim.
A.2 Global and local ields
P oblem A.2. Le Abe a comple e alua ion ing and mi s unique maximal
ideal. Then, A∼
=lim
←−nA/mnas opological ings.
P oo . We shall see he canonical map ϕ:A→lim
←−nA/mnis an isomo -
phism. I is clea ly a ing homomo phism. Thus, since he ke nel is Tn≥1mn=
0, i is injec i e. To check su jec i i y, no e ha an elemen s∈lim
←−nA/mnis
44
Appendix A. Sol ed P oblems
gi en by an in ini e uple s= (sn)whe e
sn=a0+a1π+···+an−1πn−1,
o aia e aken in a se o ep esen a i es o he cose s and πa uni o mize o
A. Thus, (sn)is no hing bu he image by ϕo he elemen Pn≥0anπn∈A.
Hence, ϕis bijec i e and an isomo phism o ings.
We a e le o see i is con inuous o i o be a homemo phism oo. I is
enough o check ha he basis o neighbo hoods mno 0 in Aa e mapped
o a basis o neighbo hoods o 0 in lim
←−nA/mn. Since he open se s Nn=
Qk≥nA/mk o m a basis o neighbo hoods o 0 in Qn≥1A/mkand ϕ(mn) =
Nn∩lim
←−nA/mn,ϕis con inuous and hus an homemo phism.
P oblem A.3. Le Abe a comple e alua ion ing and mi s unique maximal
ideal. Then, mn/mn+1 ∼
=A/m.
P oo . No e he elemen s a∈Amay be w i en as he sums
a=X
n≥0
anπn,
whe e ana e aken in a se o ep esen a i es o he cose s and πis a uni-
o mize o A. The elemen s o he ideal mna e he sums
mn=X
k≥n
akπk.
Thus, he elemen s in mn/mn+1 a e o he o m anπn+mn+1 and a e in a
clea one- o-one co espondence wi h he elemen s in he esidue ield (gi en
by he canonical epimo phism) p o ing he esul .
P oblem A.4. The esidue ield o a global ield Kis ini e.
P oo . Le i s Kbe a global unc ion ield. Then, K=Fq[ ]and i is a
p incipal ideal domain (PID) and a nonze o p ime ideal is a maximal ideal
gi en by a nonze o i educible polynomial . Then
Fq[ ]
( )={a0+a1 +···+an−1 n−1:ai∈Fq}∼
=Fqn,
whe e n= deg . Hence, he esidue ield is ini e as i is a ini e dimensional
ec o space o e a ini e ield.
Now, le Kbe a numbe ield, i.e. a ini e ex ension o Q. Then, OKis
ini e dimensional o e Z. Thus, i is enough o show ha o any posi i e
p ime in ege p, and ν= o dp, he esidue ield Oν/pis ini e whe e Oν=Z(p)
(i.e. he localiza ion o Za he p ime ideal (p)) and p=pZ(p). We shall see
Oν/p=Z(p)/pZ(p)∼
=Fp. This ollows di ec ly om he ollowing Lemma.
45
A.4. Class ield heo y
index and ini e subg oups o i s cha ac e g oup ia he assignmen s H≤G7→
ϕ(H) := {χ∈G∗:χ|H= 0}and H≤G∗7→ ψ(H) := Tχ∈Hke χ.
P oo . Fi s we see hese maps a e well de ined. Fo ha , le H≤Gbe
open o ini e index. Then, he cha ac e s in ϕ(H)a e in one- o-one co e-
spondence wi h he ones o G/H and since His o ini e index |ϕ(H)|=
|(G/H)∗|=|G:H|<∞. Now, le H≤G∗be ini e. Then, ψ(H)is mapped
in o Lχ∈HG/ ke χand since each G/ ke χis ini e and H oo, G/H is ini e
p o ing i is o ini e index. Also, i is he in e sec ion o open se s; hus, open.
Now, we shall see ψ(ϕ(H)) = Hand ϕ(ψ(H)) = H. Fi s , le H≤Gopen
o ini e index. Clea ly, ψ(ϕ(H)) ≤H. Should his inclusion no be an equali y,
he quo ien ψ(ϕ(H))/H would be non- i ial and he e would be a cha ac e
χ∈G∗such ha χ∈ϕ(H)bu χ|ψ(ϕ(H)) 6= 0, which is a con adic ion.
Now, le H≤G∗ ini e. Clea ly, H≤ϕ(ψ(H)). Now, ϕ(ψ(H)) can be
iden i ied wi h (G/ψ(H))∗and Hwi h a subg oup o i . Bu Hclea ly sepa-
a es any wo poin s in G/ψ(H), bu no p ope subg oup o (G/ψ(H))∗does
so; hus, Hcanno be p ope and we ge he equali y H=ϕ(ψ(H)).
52
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