Integral bases and monogenity of pure fields

In eg al bases and monogeni y o pu e ields Is ´an Ga´al∗ , and L´aszl´o Reme e Uni e si y o Deb ecen, Ma hema ical Ins i u e H–4002 Deb ecen P .400., Hunga y e–mail: gaal.is [email p o ec ed]u, [email p o ec ed]u Ap il 26, 2016 Abs ac Le mbe a squa e- ee in ege (m6= 0,1). We show ha he s uc- u e o he in eg al bases o he ields K=Q(n √m) a e pe iodic in m. Fo 3 ≤n≤9 we show ha he pe iod leng h is n2. We explici ly de- sc ibe he in eg al bases, and o n= 3,4,5,6,8 we explici ly calcula e he index o ms o K. This enables us in many cases o cha ac e ize he monogeni y o hese ields. Using he explici o m o he index o ms yields a new echnics ha enables us o de i e new esul s on monogeni y and o ge se e al o me esul s as easy consequences. Fo n= 4,6,8 we gi e an almos comple e cha ac e iza ion o he monogeni y o pu e ields. 1 In oduc ion Le mbe a squa e- ee in ege (m6= 0,1) and n≥2 a posi i e in ege . The e is an ex ensi e li e a u e o pu e ields o ype K=Q(n √m). (Desc ibing he ∗Resea ch suppo ed in pa by K115479 om he Hunga ian Na ional Founda ion o Scien i ic Resea ch 2010 Ma hema ics Subjec Classi ica ion: P ima y 11R04; Seconda y 11Y50 Key wo ds and ph ases: pu e ields, in eg al basis, powe in eg al basis, monogeni y ollowing esul s on pu e ields we use some basic concep s on monogeni y and powe in eg al bases ha a e de ailed in Sec ion 2.) B.K.Spea man and K.S.Williams [14] ga e an explici o mula o he in eg al basis o pu e cubic ields. B.K.Spea man, Y.Qiduan and J.Yoo [13] showed ha i iis a cube ee posi i e in ege hen he e exis in ini ely many pu e cubic ields wi h minimal index equal o i. I.Ga´al and T.Szab´o [11] s udied he beha iou o he minimal indices o pu e cubic ields in e ms o he disc iminan . L. El Fadil [12] ga e condi ions o he exis ence o powe in eg al bases o pu e cubic ields in e ms o he index o m equa ion. T.Funaku a [7] s udied he in eg al basis in pu e qua ic ields. I.Ga´al and L.Reme e [9] calcula ed elemen s o index 1 (wi h coe icien s <101000) in pu e qua ic ields K=Q(4 √m) o 1 < m < 107,m≡2,3 (mod 4). S.Ahmad, T.Nakaha a and S.M.Husnine [3] showed ha i m≡1 (mod 4), m6≡ ±1 (mod 9) hen Q(6 √m) is no monogenic. On he o he hand [4], i m≡2,3 (mod 4), m6≡ ±1 (mod 9) hen Q(6 √m) is monogenic. A.Hameed and T.Nakaha a [1] cons uc ed in eg al bases o pu e oc ic ields Q(8 √m). They p o ed [2] ha i m≡1 (mod 4) hen Q(8 √m) is no monogenic. On he o he hand A.Hameed, T.Nakaha a, S.M.Husnine and S.Ahmad [5] p o ed ha i m≡2,3 (mod 4) hen Q(8 √m) is monogenic. A.Hameed, T.Nakaha a, S.M.Husnine and S.Ahmad [5] showed ha i m≡2,3 (mod 4) hen Q(2n √m) is monogenic, his in ol es he pu e qua ic and pu e oc ic ields, as well. Mo eo e , hey showed [5] ha i all he p ime ac o s o ndi ide m hen Q(n √m) is monogenic. Ou pu pose is o 3 ≤n≤9 o gi e a gene al cha ac e iza ion o he in eg al basis o K=Q(n √m). We p o e ha he in eg al bases o K= Q(n √m) is pe iodic in m. Fo 3 ≤n≤9 he pe iod leng h is n2. The knowledge o he in eg al bases makes possible also o compe e he spo adic esul s on he monogeni y o hese ields. Ou me hod applying he explici o m o he index o ms yields a new echnics ha enables us o ob ain new esul s on he monogeni y o hese ields and o ob ain se e al o me esul s as easy consequences. In ou Theo ems 4, 7, 8 we gi e an almos comple e cha ac e iza ion o he monogeni y pu e qua ic, sex ic and oc ic ields, espec i ely. The cubic case is well-known and easy, much less is known abou he quin ic, sep ic and nonic cases. 2 2 Basic concep s abou he monogeni y o num- be ields We ecall hose concep s [8] ha we use h oughou . Le αbe a p imi i e in eg al elemen o he numbe ield K( ha is K=Q(α)) o deg ee nwi h ing o in ege s ZK. The index o αis I(α) = (Z+ K:Z[α]+) = s D(α) DK =1 p|DK|Y 1≤i<j≤nα(i)−α(j), whe e DKis he disc iminan o Kand α(i)deno e he conjuga es o α. The minimal index o Kis iK= min I(α) whe e α uns h ough he p imi i e in eg al elemen s o K. I B= (b1= 1, b2, . . . , bn) is an in eg al basis o K, hen he index o m co esponding o his in eg al basis is I(x2, . . . , xn) = 1 p|DK|Y 1≤i<j≤n(b(i) 2−b(j) 2)x2+. . . + (b(i) n−b(j) n)xn (whe e b(i) jdeno e he conjuga es o bj) which is a homogeneous polynomial wi h in eg al coe icien s. Fo he in eg al elemen α=x1+b2x2+. . . +bnxn we ha e I(α) = |I(x2, . . . , xn)| independen ly o x1.αgene a es a powe in eg al basis (1, α, . . . , αn−1) i and only i I(α) = 1 ha is (x2, . . . , xn)∈Zn−1is a solu ion o he index o m equa ion I(x2, . . . , xn) = ±1 in(x2, . . . , xn)∈Zn−1.(1) In his case ZK=Z[α] and Kis called monogenic. 3 3 Basic esul s Th oughou we assume ha mis a squa e- ee in ege wi h m6= 0,1 and n > 2 an in ege . Le K=Q(n √m) and ϑ=n √m. Ou i s heo em is on he p ime di iso s o he denomina o s o he in eg al basis elemen s: Theo em 1. I (1, ϑ, . . . , ϑn−1)is no an in eg al basis in K, hen o any elemen α=a0+a1ϑ+. . . +an−1ϑn−1 q(2) o he in eg al basis (wi h a0, . . . , an−1, q ∈Z,q6= 0) he denomina o qcan only be di isible by p imes di iding n, he p ime ac o s o qdo no di ide m. P oo The disc iminan o ϑ=n √mis ±nnmn−1. I (1, ϑ, . . . , ϑn−1) is no an in eg al basis in Q(ϑ), hen he e mus be a numbe qdi iding nnmn−1and an elemen αo ype (2) such ha αis an algeb aic in ege and an elemen o (1, ϑ, . . . , ϑn−1) can be eplaced by α o ge a basis wi h smalle disc iminan . Le pbe a p ime di iso o q. Then ob iously α0=q pα=e0+e1ϑ+. . . +en−1ϑn−1 p(3) is also an algeb aic in ege . We can also assume ha 0 ≤ei< p (0 ≤i≤ n−1) by aking each eimodulo p. We show ha pis a di iso o n. Assume on he con a y ha p|m. The elemen α0ϑ=e0ϑ+e1ϑ2+. . . +en−2ϑn−1+en−1m p is ob iously an algeb aic in ege . By p|m he elemen e0ϑ+e1ϑ2+. . . +en−2ϑn−1 p 4 is also an algeb aic in ege . We p oceed by mul iplying his elemen by ϑ and omi ing he analogous in eg al pa . Finally we ob ain ha %=e0ϑn−1 p is an algeb aic in ege . The elemen %is he oo o he polynomial %(x) = pxn−en 0m pn−1 . This polynomial is i educible o e Qi and only i i s ecip ocal polynomial 1/%(x) = en 0m pn−1 xn−p. is i educible. He e m/p is an in ege , no di isible by pbecause mis squa e- ee. e0is also no di isible by p(o he wise we did no ha e e0in (3) and we had he same esul wi h he i s non-ze o ei). Hence 1/%(x) is an Eisen- ein polynomial, he e o e %(x) is i educible. Then %(x) is he de ining polynomial o %. This con adic s o %being an algeb aic in ege . 2 Rema k. Theo em 1 implies Theo em 3.1. o A.Hameed, T.Nakaha a, S.M.Husnine and S.Ahmad [5]: i all he p ime ac o s o ndi ide m hen Q(n √m) is monogenic. Nex we show ha he in eg al bases o K=Q(n √m) a e pe iodic. Fi s we p o e his s a emen wi h a pe iod leng h much la ge han n2bu his esul is alid o any n. Theo em 2. Le n=ph1 1. . . phk kand n0=p[nh1/2] 1. . . p[nhk/2] kwhe e [x]deno es he lowe in ege pa o x. Le ϑ=n √mand γ=n √m+nn 0. Then he s uc u e o he in eg al bases o he ields Q(ϑ)and Q(γ)is he same in e ms o ϑand γ, espec i ely. Rema k. Unde he ”same s uc u e” we mean ha i he in eg al basis o Q(ϑ) has an elemen a0+a1ϑ+. . . +an−1ϑn−1 q, 5 hen he in eg al basis o Q(γ) has an elemen a0+a1γ+. . . +an−1γn−1 q and ice e sa. P oo Assume ha (1, ϑ, . . . , ϑn−1) is no an in eg al basis in Q(ϑ). Then he e mus be in ege elemen s o ype α=a0+a1ϑ+. . . +an−1ϑn−1 q(4) which can eplace elemen s o (1, ϑ, . . . , ϑn−1) o ob ain an in eg al basis. We show ha he exis ence o analogous algeb aic in ege s o ype (4) is equi alen in he ields gene a ed by ϑ=n √mand by γ=n √m+nn 0. I we eplace an elemen o he basis (1, ϑ, . . . , ϑn−1) by αo (4) hen he disc iminan o he basis dec eases by a ac o q2. Hence q2di ides ±nnmn−1 ( he disc iminan o ϑ=n √mis ±nnmn−1). By Theo em 1 he p ime di iso s po qdo no di ide m. Hence q2|nn, which implies ha qdi ides n0. Deno e he conjuga es o αby α(j), j = 1, . . . , n. The de ining polynomial o αis n Y j=1 (x−α(j)) = 1 qn n Y j=1 (qx −a0−a1ϑ(j)−. . . −an−1(ϑ(j))n−1). The p oduc is a symme ical polynomial o ϑ(1), . . . , ϑ(n), hence i s coe - icien s can be exp essed as polynomials (wi h in ege coe icien s) o he de ining polynomial o ϑ, ha is xn−m. Hence he e exis polynomials P0, . . . , Pn−1∈Z[x] such ha n Y j=1 (x−α(j)) = 1 qn((qx)n+Pn−1(m)(qx)n−1+. . . +P1(m)(qx) + P0(m)). The e o e he elemen αis an algeb aic in ege i and only i qn|qjPj(m) ha is qn−j|Pj(m) (j= 0,1, . . . , n −1).(5) 6 Replace now mby m0=m+nn 0and conside in eg al bases in he ield Q(γ) = Q(n √m+nn 0). In his ield an elemen o ype (4), ha is δ=a0+a1γ+. . . +an−1γn−1 q(6) is an algeb aic in ege i and only i qn−j|Pj(m+nn 0) (j= 0,1...,n−1) (7) wi h he same polynomials Pj. By q|n0 he condi ions (5) a e equi alen o (7). The e o e Q(ϑ) and Q(γ) con ains he same ype o in ege elemen s. Elemen s o ha ype a e linea ly independen in he i s case i and only i hey a e linea ly independen in he second case. The e o e Q(ϑ) and Q(γ) admi s he same ype o in eg al bases. 2 4 The s uc u e o he in eg al bases is pe iodic in mmodulo n2 o 3≤n≤9 Theo em 2 implies ha he in eg al bases o K=Q(n √m) a e pe iodic mod- ulo nn 0. This numbe is o magni ude nn2/2. Fo small alues o nwe ha e a much sha pe asse ion. Theo em 3. Fo 3≤n≤9 he in eg al bases o Q(n √m)a e pe iodic in m modulo n2. P oo Fo n= 3,4,5 he nn 0is 27, 655536, 9765625, espec i ely. Calcula ing he in eg al bases o Q(n √m) o squa e- ee mup o nn 0i is easily seen ha he s uc u e o he in eg al bases o Q(n √m) a e pe iodic modulo n2. One can easily de ec a ew ypes o in eg al bases ha a e epea ed o squa e- ee alues o m,m+n2,m+ 2n2e c. Le now n > 5. Then nn 0is a oo la ge o he calcula ions desc ibed abo e. Howe e o n= 6,7,8,9 we managed o p o e he same asse ion. 7 Le 1 < < n2. I is squa e- ee, hen se 0= . I has a common squa e ac o wi h n, hen none o +kn2is squa e- ee, we omi . I has no common squa e ac o wi h nbu con ains ano he squa e ac o , hen we se 0= +n2o 0= + 2n2e c. which is al eady squa e- ee. Le ϑ=n √ 0, calcula e he in eg al bases o Q(n √ 0) and deno e he basis elemen s by (b1= 1, b2, . . . , bn), whe e bjis o he o m bj=aj0+aj1ϑ+. . . +aj,n−1ϑn−1 q (wi h aj0, aj1, . . . , aj,n−1∈Zand wi h a non-ze o denomina o q he p ime ac o s o which di ide n). Le m= +kn2be a squa e- ee in ege , γ=n √mand b0 j=aj0+aj1γ+. . . +aj,n−1γn−1 q. We wonde I. i he analogues o he elemen s bj, ha is he elemen s b0 j emain algeb aic in ege o any squa e- ee m= +kn2, u he II. i o some squa e- ee m= +kn2some o he basis elemen s (b0 1= 1, b0 2, . . . , b0 n) can be eplaced by an in eg al elemen o ype d=e1b0 1+. . . +enb0 n p(8) (whe e 0 ≤e1, . . . , en≤p−1 and pis a p ime di iso o n) o ob ain a basis wi h smalle disc iminan . I. The de ining polynomial o e1b0 1+. . . +enb0 nis G(x) = n Y j=1 (x−e1b0(j) 1−. . . −enb0(j) n). This polynomial is symme ical in he conjuga es o γ=n √m, hence i s coe icien s will be polynomials in m: G(x) = xn+Gn−1(m)xn−1+. . . +G1(m)x+G0(m). 8 Since he e a e denomina o s in he b0 i, he polynomials Gj(depending also on e1, . . . , en) a e no necessa ily o in ege coe icien s. Le us subs i u e m= +kn2. We ob ain G(x) = xn+Hn−1(k)xn−1+. . . +H1(k)x+H0(k). Fo all possible esidues we ha e explici ly calcula ed hese polynomials Hj(k) which also depend on e1, . . . , en. In all cases we ound ha hese a e polynomials in k, e1, . . . , enwi h in ege coe icien s. Subs i u ing ei= 1 and ej= 0, j = 1, . . . , n, j 6=i his implies ha b0 iis in ege o any squa e- ee m= +kn2(1 ≤i≤n). II.Conside now he de ining polynomial o d(8), ha is P(x) = 1 pnG(px) = 1 pn(px)n+Hn−1(k)(px)n−1+. . . +H1(k)(px) + H0(k). This polynomial has in ege coe icien s i and only i pndi ides pjHj(k) ha is pn−j|Hj(k) = Hj(k, e1, . . . , en) (0 ≤j≤n−1).(9) Le now (e1, . . . , en)∈Znbe an a bi a y gi en ixed uple. Ob iously by (9) he alidi y o he s a emen i dis in eg al o no , depends only on he beha iou o kmodulo pnand no on he alue o k. This allows us o es he ields Q(n √m) o squa e- ee m= +kn2whe e k uns hough all esidue classes modulo pn. These ields we e es ed di ec ly, calcula ing hei in eg al bases. We ound ha in all cases he ields had he same s uc u e o in eg al basis in e ms o γ=n √mlike he ield Q(n √ 0) in e ms o ϑ=n √ 0. This p o es ou asse ion. 2 Rema k The es desc ibed a he end o he abo e p oo equi ed o calcu- la e he in eg al bases o 24(26+ 36) = 18239 sex ic ields, 48 ·77= 39530064 sep ic ields, 48 ·28= 12288 oc ic ields and 72 ·39= 1417176 nonic ields. 9 Case 6.3. = 10,19, m= + 36ksqua e- ee 1, x, x2, x3,1 + x2+x4 3,x+x3+x5 3, D = 2632m5 Calcula ing he index o m i is easily seen ha he index o m equa ion is no sol able modulo 3, hence hese ields a e no monogenic. This case is no co e ed by S. Ahmad, T. Nakaha a and S. M. Husnine [3], [4] since m≡1 (mod 9). Case 6.4. = 26,35, m= + 36ksqua e- ee 1, x, x2, x3,1+2x2+x4 3,x+ 2x3+x5 3, D = 2632m5 This case is no co e ed by S. Ahmad, T. Nakaha a and S. M. Husnine [3], [4] since m≡ −1 (mod 9). I m= 26+36k hen 4m|( 2−9 3). I Kis monogenic hen o a solu ion o he index o m equa ion hese ac o s a e equal o ±1. The possible alues o 2−9 3a e ±8,±10, hence he abo e di isibili y can no hold. I m= 35 + 36k hen 4(35 + 36k)|( 2−9 3). I Kis monogenic hen o a solu ion o he index o m equa ion hese ac o s a e equal o ±1. The possible alues o 2−9 3a e ±8,±10, hence he abo e di isibili y can only hold o k=−1, ha is m=−1. I is easily seen ha he ela i e index [10] o elemen s o K=Q(6 √−1) is di isible by 9, he e o e his ield is no monogenic, ei he . Case 6.5. = 17, m= + 36ksqua e- ee 1, x, x2,1 + x3 2,4+3x+ 2x2+x4 6,4x+ 3x2+ 2x3+x5 6, D = 32m5 This case is no co e ed by S. Ahmad, T. Nakaha a and S. M. Husnine [3], [4] since m≡ −1 ( mod 9). Calcula ing he index o m we can easily see ha he index o m equa ion is no sol able modulo 6. Case 6.6. m= 1, m= 1 + 36ksqua e- ee 1, x, x2,1 + x3 2,4+3x+ 4x2+x4 6,3+4x+ 3x2+x3+x5 6, D = 32m5 16 This case is no co e ed by S. Ahmad, T. Nakaha a and S. M. Husnine [3], [4] since m≡1 (mod 9). Calcula ing he index o m we can easily see ha he index o m equa ion is no sol able modulo 3. Summa izing he abo e s a emen s we ha e Theo em 7. Fo he ollowing alues o le m= + 36k(k∈Z) be a squa e- ee in ege . The ield K=Q(6 √m)is monogenic o = 2,3,6,7,11,14,15,22,23,30,31,34 and is no monogenic o = 1,5,10,13,17,19,21,25,26,29,33,35. 6 Pu e sep ic ields, K=Q(7 √m) Case 7.1. = 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,20,21,22,23,24, 25,26,27,28,29,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47, m= + 49ksqua e- ee B={1, x, x2, x3, x4, x5, x6}, D =−m677 These ields a e ob iously monogenic. Case 7.2. = 18, m= 18 + 49ksqua e- ee B={1, x, x2, x3, x4, x5,1+2x+ 4x2+x3+ 2x4+ 4x5+x6 7}, D =−m675 Case 7.3. = 19, m= 19 + 49ksqua e- ee B={1, x, x2, x3, x4, x5,1+3x+ 2x2+ 6x3+ 4x4+ 5x5+x6 7}, D =−m675 17 Case 7.4. = 30, m= 30 + 49ksqua e- ee B={1, x, x2, x3, x4, x5,1+4x+ 2x2+x3+ 4x4+ 2x5+x6 7}, D =−m675 Case 7.5. = 31, m= 31 + 49ksqua e- ee B={1, x, x2, x3, x4, x5,1+5x+ 4x2+ 6x3+ 2x4+ 3x5+x6 7}, D =−m675 Case 7.6. = 48, m= 48 + 49ksqua e- ee B={1, x, x2, x3, x4, x5,1+6x+x2+ 6x3+x4+ 6x5+x6 7}, D =−m675 Case 7.7. = 1, m= 1 + 49ksqua e- ee B={1, x, x2, x3, x4, x5,1 + x+x2+x3+x4+x5+x6 7}, D =−m675 7 Pu e oc ic ields, K=Q(8 √m) In all hese cases he index o m is he p oduc o h ee ac o s o deg ees 4,8,16, espec i ely. We shall deno e hese ac o s by 1, 2, 3. These depend on he pa ame e mand on he a iables x2, . . . , x8. Case 8.1. = 2,3,6,7,10,11,14,15,18,19,22,23,26,27,30,31,34,35,38, 39,42,43,46,47,50,51,54,55,58,59,62,63, m= + 64ksqua e- ee 1, x, x2, x3, x4, x5, x6, x7, x8, D =−88m7 18 These ields a e ob iously monogenic. This also ollows om A.Hameed, T.Nakaha a, S.M.Husnine and S.Ahmad [5]. Case 8.2. = 1,17,33,49, m= + 64ksqua e- ee 1, x, x2, x3,1 + x4 2,x+x5 2,1 + x2+x4+x6 4, 1 + x+x2+x3+x4+x5+x6+x7 8, D =−210m7 These ields a e no monogenic by he heo em o A.Hameed and T.Nakaha a [2]. We conjec u e ha in hese ields he minimal index is 128. Case 8.3. = 5,13,21,29,37,45,53,61, hese cases can be included by m= 5 + 8k, squa e- ee 1, x, x2, x3,1 + x4 2,x+x5 2,x2+x6 2,x3+x7 2, D =−216m7 Calcula ing and ac o izing 3−16 2 2we ind ha i is di isible by m. I he e exis ed a powe in eg al basis hen o a solu ion o he index o m equa ion we would ha e 1, 2, 3=±1, hence 3−16 2 2=±1−16 is ei he −15 o −17. The possible di iso s a e ±3,±5,±15,±17 bu only m=−3,5 is o ype m= 5 + 8k. Fo m=−3 hen elemen (−1,−1,0,1,1,0,−1) has index one, hence K=Q(8 √−3) is monogenic. Fo m= 5 he leas index we ound in K=Q(8 √5) was 16. No e ha A.Hameed, T.Nakaha a, S.M.Husnine and S.Ahmad [2] asse ha hese ields a e no monogenic, hey ce ainly did no in ol e K= Q(8 √−3). Case 8.4. = 9,25,41,57, hese cases can be included by m= 9 + 16k, squa e- ee 1, x, x2, x3,1 + x4 2,x+x5 2,1 + x2+x4+x6 4,x+x3+x5+x7 4, D =−212m7 19 Calcula ing and ac o izing 2−4 2 1we ind ha i is di isible by m. I he e exis ed a powe in eg al basis hen o a solu ion o he index o m equa ion we would ha e 1, 2, 3=±1, hence 2−4 2 1=±1−4 is ei he −3 o −5. The possible di iso s a e ±3,±5 bu none o hem is o ype m= 9 + 16k. The e o e hese ields a e no monogenic. This also ollows om he heo em o A.Hameed, T.Nakaha a, S.M.Husnine and S.Ahmad [2]. Summa izing he abo e s a emen s we ha e Theo em 8. Fo he ollowing alues o le m= + 64k(k∈Z) be a squa e- ee in ege . The ield K=Q(8 √m)is monogenic o = 2,3,6,7,10,11,14,15,18,19,22,23,26,27,30,31,34,35,38,39,42,43,46, 47,50,51,54,55,58,59,62,63 and is no monogenic o = 1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,m6= 5, wi h he ex- cep ion o K=Q(8 √−3) which is monogenic. Rema k 9. We conjec u e ha he minimal index o K=Q(8 √5) is 16. Oc ic ields o his ype will be conside ed in a o ecoming pape . 8 Pu e nonic ields, K=Q(9 √m) Case 9.1. = 2,3,4,5,6,7,11,12,13,14,15,16,20,21,22,23,24,25,29,30, 31,32,33,34,38,39,40,41,42,43,47,48,49,50,51,52,56,57,58,59,60,61,65, 66,67,68,69,70,74,75,76,77,78,79, m= + 81k, squa e- ee 1, x, x2, x3, x4, x5, x6, x7, x8, D = 318m8 These ields a e ob iously monogenic. Case 9.2. = 1,28,55, m= + 81k, squa e- ee 1, x, x2, x3, x4, x5,1 + x3+x6 3,x+x4+x7 3, 20 1 + x+x2+x3+x4+x5+x6+x7+x8 9, D = 310m8 Case 9.3. = 8,17,35,44,62,71, m= + 81k, squa e- ee 1, x, x2, x3, x4, x5,1+2x3+x6 3,x+ 2x4+x7 3,x2+ 2x5+x8 3, D = 312m8 Case 9.4. = 10,19,37,46,64,73, m= + 81k, squa e- ee 1, x, x2, x3, x4, x5,1 + x3+x6 3,x+x4+x7 3,x2+x5+x8 3, D = 312m8 Case 9.5. = 26,53,80, m= + 81k, squa e- ee 1, x, x2, x3, x4, x5,1+2x3+x6 3,x+ 2x4+x7 3, 1+2x+x2+ 8x3+ 7x4+ 8x5+x6+ 2x7+x8 9, D = 310m8 9 Compu a ional ema ks In all ou calcula ions we used Maple [6] and mos o ou p og ams execu ed a couple o seconds o a ew minu es on an a e age lap op. Fo n= 4,6,8 we needed a e y ca e ul calcula ion o he ac o s o he index o ms, which may ake ex emely long o he wise. The es s co esponding o Theo em 3 ook also a ew minu es o n= 3,4,5,6,8. Fo n= 9 i execu ed 5 hou s. Fo n= 7 we execu ed ou Malpe p og am on a supe compu e wi h nodes ha ing 24 CPU-s. 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