A primitivity criterion

Pub . Ma . UAB N 0 20 Se . 1980 Ac es VII JMHL A PRIMITIVITY CRITERION En ic Na , Nú iaVila Seccióde Ma emá iques Uni e si a Au bnoma de Ba celona Abs ac . In his no e we gi e a gene aliza ion o Fu wáng1e 's p imi i- i y c i e ion [21, in o de o assu e ha a polynomial is p imi i e h ough his coe icien s . Le K be a ield . We ecall ha a polynomial (X)'EK[ X]  is called p imi i e o e K i i s Galois a oup o e K is p imi i e as a pe mu a ion g oup o i s oo s [3,ch .VI,491 . Th oughou hisno e R willdeno e a Dedekinddomain and K i s ield o quo ien s . I A is a p ime ideal o R, we deno eby ,~ he alua ion o R associa ed o ~ . Fu wángle p o ed he ollowing c i e ion [2, h .31 : I (X)=X n +a 1 X n-l + . . .+a n EIL[X1 is an i educible polynomial and o a p ime p is p (a i »0, 1<i-<n, p (a n-1)=1 and p (a n )>1, hen (X) is p imi i e . In his no e we p o e he ollowing gene aliza ion : Theo em . Le (X)=X n +a 1 X n-1 + . . .+a n ER[X1 be an i educible poly- nomial . Le 1 be a p ime ideal o R such ha e i = 1 (a i »l o e e y 1<i-,Qn . Le OQc<n be such ha el /i ? ek /k o e e y 1<i-<n . Suppose ha n= s, and he oo so (X) can be di ided in s subse s o imp imi i i y . I in e e ys- uple (ii, . . .,is) o in- dexs wi h 0<i m < , lZ-'ín<,s, andil+ . . .+is=k, he eexis san index i q such ha (i q ,k)=1, hen s > k/(k,ek) . 186 Fi s we need an easylemma : Lemma . Le (X)=X n +a 1 X n-1 + . . .+a n ER[X] . Le a be a Le Ibe a p ime idealo K .and p a n ime ideal o K(a) lying o e ,V .  Le ' XEQand e=e ( /V . i) I  (ai )>iX  o e e y  hen 1,(«»eX ii)  I ,, (a : )>i> ,  o e e y himen, hen ,~ (a)>e?, P oo . The slope o any segmen o he New on's polygon asso- cia ed o (X)  is  > X ,  by  [1 ,ch .2,51  is 1,(a)/e>k . P oo o he heo em . Le L be a spli ing ieldo (X) o e Le 12 be a p ime ideal o L lying o e ng,and e=e(p,A ) . Le 1  1 2  2  s al, . . .,a ;al, . . .,a ; . . . ;as, . ... a . K . be Le a di ision o he oo s o (X) in subse so imp imi i i y . i (x) = j1il (x-a~) = x +l1x -1+ . . .+ , l , -!~i<-s . Clea ly he elemen s 11, . ... s a e conjuga ed o e K o 1<j<- . Le Thus, ,(bi) g j (X) =XS +biX s-1 + . . .+bs, l<j< , oo o (X) . e e y be hei i educiblepolynomial o e K . Being he oo s o (X) in ege so e K, he same happens wi h he ú's, hence g j (X)E R ; ix] o e e y 15j< . I a is a oo o (X), i ollows om he lemma ha p (a)> ee k /k, ,( 1) 1 (bi) a k = 09 , ~ im , I,<Qs il+ . . .+is=k hence ijee k /k, hence s Clea ly (X)  = ¡ L l  i (X),  hence 1 s i ... i , whe e i p =1 o e e y 1<i-<-s . i s By (1) e e ysummand has ,~( . . . . i ) > ee k .  (3) 1s Since 12(ak)=eek,  he eexis s one s- uple  (¡ l o... i S ) equali y holds . i n (3) . Hence, o his s- uplewe ha e ( ¿ ) = - i m ee k /k, o e e y l~n-s ~  . m o which Le iq be he índex in his s- uple such ha (i q ,k)=1 . By (2) and ii) o he lemma, he eexis san índex , l<- <_s, such ha i ,, & (b q )  = igek/k . i Since V Y (b q)  is an in ege and (i q ,k)=1 we conclude ha e k /k is an in ege , hence is a mul iple o k/(k,e k ) . Thus s> >k/(k,ek) . he In he ollowinacases (X)  is n imi i e : i)  I k=n-1 and (n-l,en-1)=1, ii) ' I n>3, k=n-1 and e n-l =1 o 2 . iii) I n is odd, k=n-2 and (n-2,en-2)=1 . i )  I n>6, 34n, k=n-3 and (n-3,en-3)=1 . )  I p is a p imenumbe n/2 < p < n andk=p . P oo . Alla e an easy consequence o he heo em . .Le us ema k ha he e alwaysexis s a p imenumbe sa is ying he condi ion o ) by a heo em o Tchebysche . Rema k . Fu wángle 's p imi i i y c i e ion is he special case e n-l =1 in i) o he co olla y . Re e ences . 1 . E .A in, Algeb aic ' numbe s and al eb aic unc ions, Go don and B each N .Yo k,1967 . 2 . Ph . Fu wángle , Ube K i e ien ü i eduzible und ü p imi i- e Gleichungen und übe die Au s ellung a ek eie Gleichun- gen, Ma h . Ann . 85 (1922) 34-40 . 3 . B .L . Van de Wae den,Mode nAlgeb a, Vol .I, Unga , N .Yo k,1953 .