Mathematica Slovaca. Antonín Dvořák; David Jedelský; Juraj Kostra The fields of degree seven over rationals with a normal basis generated by a unit

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1 Mathematica Slovaca Antonín Dvořák; David Jedelský; Juraj Kostra The fields of degree seven over rationals with a normal basis generated by a unit Mathematica Slovaca, Vol. 49 (1999), No. 2, Persistent URL: Terms of use: Mathematical Institute of the Slovak Academy of Sciences, 1999 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 Math. SlOVaCa, 49 (1999), NO. 2, r\/tathematica Slovaca 1999 Slovák Academy of Sciences THE FIELDS OF DEGREE SEVEN OVER RATIONALS WITH A NORMAL BASIS GENERATED BY A UNIT ANTONIN DVORAK* DAVID JEDELSKY** JURAJ KOSTRA*** (Communicated by Stanislav Jakubec ) ABSTRACT. In this paper we determine all fields of degree seven over rationals which have an integral normal basis generated by a unit. In this paper we determine all fields of degree 7 over rationals which have integral normal basis generated by a unit. In [1] the following necessary condition was found for the existence of such a basis in the prime extensions of rationals. THEOREM 1. Let K be a tamely ramified extension of Q of prime degree I and let m=p l -p 2...p 3 be a conductor of the field K. Let there exist an integral normal basis generated by a unit in the field K. Then or l l = 1 (modpj, i = 1,2,...,$, l l = -1 (mod p.), i = 1,2,..., 5. Let [K : Q] = 7 with the conductor m. If there exists an integral normal basis generated by a unit in the field K, then the above theorem determines the possible values of m. We have to find all primes p, p = 1 (mod 7), for which 7 7 = 1 (mod p), and all primes p, p = 1 (mod 7), for which 7 7 = -1 (modp). From the first congruence we get three possible values of the conductor m: m = 29, m = 4733, m = 29 * AMS Subject Classification (1991): Primary 11R33. Key words: normal basis, unit, circulant matrix. This paper has been supported by grant 201/97/0433 of GA CR 143

3 ANTONÍN DVOŘÁK DAVID JEDELSKÝ JURAJ KOSTRA From the second congruence we get the three remaining possible values of the conductor m: m=113, m = 911, m = 113* 911. Let m be a prime number. Then there exists only one field K, [K : Q] = 7, with Galois group G(Q(( m )/K) generated by cr 11^-, where a is the generator ofo(q(cro)/*0- Let m = PiP 2 be a product of two primes p x and p 2, then, by [2; Lemma 2], an arbitrary K C Q(C m )' [If : Q] = 7, is a subfield of the composite L K X K 2 where [K x : Q] = 7, [K 2 : Q] = 7 and K x C Q(C P1 ), K 2 C Q(C P2 ) The Galois group is G(K X K 2 /Q) = G(K X /Q) x G(K 2 /Q), where G(KJQ) is generated by p and G(K 2 /Q) is generated by r. In this situation, there exist six fields K with conductor m and [K : Q] =7. They are determined by the subgroups G(L/K) of G(L/Q) generated by 1. [P,T] 4. [p,t 4 ] 2. [p,t 2 ] 5. [p,t 5 ] 3. [p,t 3 ] 6. [p,t 6 ]. In any such field K, there exists an integral normal basis generated by a = Tr Q(Cm)/K(Cm)» where C m 1s a primitive mth root of unity. We have the following possibilities for the fields K: (1) For m = 29, Q C K x C Q(C 29 ), N Ki/Q (a) = -1. (2) For m = 4733, Q C K 2 C Q(C A733 ), N K2/Q (a) = (3) For m = 29*4733, fields K 3,...,K S correspond to subgroups of G(L/K) generated by the restrictions of automorphism S G G f (Q(C m )/Q) > i n *^e following table: к i 5 NK t /Q( a ) Ґ i_ү /85537 S>ro ""*' <>m C ^m ^yč ^m ґ ^v z ^m ^ S m C н>c ^m ^m C ь->c ^ГО 'ГО Ґ,_. ЛШ260 <> m ^ S,m

4 THE FIELDS OF DEGREE SEVEN OVER RATIONALS (1) For m = 113, Q C K 9 C Q(C 113 ), N Kg/Q (a) = l. (2) For m = 911, Q C K 10 C Q(C 9 n), N Klo/q (a) = (3) For m = 113 * 911, the fields K n,...,k 16 correspond to subgroups of G(L/K) generated by the restrictions of the automorphism S G(Q(Cm)/Q). in the following table: \ к * S C m "+ C m (._. /-5134 ^m ^ S m C ^m "> ^m Ci 8413 Cn ^m " ^m Clľ 81 ґ ң. ^50673 ^m ^m /-., / *»m ^ S m - /.-,/<_(«) For iifj (ra = 29) and K 9 (m = 113) the generator a of the normal basis is a unit. For the remaining K { we shall investigate the set of all integral normal bases as follows: We shall use the fact that, in our case, the linear mapping which transforms an integral normal basis of the field K into an integral normal basis of the field K is represented by a unimodular circulant matrix of degree 7. By [1] the group of all unimodular circulant matrices of degree 7 is isomorphic to the subgroup E of finite index in the group of all units of the field Q(C 7 ) For the subgroup E we have E = {7 e Q(C 7 ); 7 is a unit, 7 = ±1 (mod (1 - C 7 ))}. The group of all units of the field Q(C 7 ) is generated by two fundamental units, e.g. and The group E generated by and 77І = i + C 7 + C c C 7 5 ^ = i + C C c 7 5 7?i = -^W 2 _=- 1 + c 7 + C6 145

5 ANTONIN DVOftAK DAVID JEDELSKY JURAJ KOSTRA is of index 3 in the group of all units of the field Q(C 7 )- We obtain an arbitrary integral normal basis of the field K by the transformation of an integral normal basis generated by a by means of a unimodular circulant matrix which corresponds to E. Let corresponds to rj x and A = circ 7 (-l, 1,0,0,0,0,1) B = circ 7 (-l, 2,0, -1,0,2, -1) corresponds to r) 2. That means that any integral normal basis /?-., /? 2,,P? of the field K (up to sign and order) can be obtained as follows (/?!, P 2,..., P 7 ) = (a,, a 2,..., a 7 ) A* B". Now we shall investigate the set {^K/Q(PI)> (P 1,P 2,...,P 7 ) = (a 1,a 2,...,a 7 )-A'-.B% t,v Z} (*) modulo p {, where p { is appropriate prime number. If there exists an integral normal basis generated by a unit P x in the field K, clearly N K^(P X ) = ±1 (mod p.). Let the period of the matrix M modulo p i be denoted as r, i.e. we have the following: M k mod ft = l', k = r, where I' is either the unit matrix or the unit matrix with permuted rows. Let the norm N K/Q (P 1 ) = p (mod pj, where p is either 1 or -1, for some i i } y = v x from (*). If there exists an integral normal basis generated by a unit in the field K then there should exist powers i t and v t such that for some ken i t = i k ' r tr +i i) modr p t v t = ( k ' r p 1 +u i) modr p B t > (o) N K/Q(PI) = P Mft)- If such powers do not exist then no integral normal basis generated by unit exists in the field. Prime numbers Pi,p 2,... were chosen such that the periods r^, r^, i = 2, 3,... are multiples of periods r^, r^, respectively. The prime numbers which were used for computations are shown in following table. 146

6 THE FIELDS OF DEGREE SEVEN OVER RATIONALS І Pi Pi PІ *42 3* *42 5* *42 21*14 The results of the computations are in the tables at the end of this paper. The columns p i contain those primes which satisfy (o). The Tables show that in none of the fields K 2,..., K s, K l0,..., K 16 does there exist an integral normal basis generated by a unit. Thus the integral normal basis generated by a unit exists only in the fields K 1 and K 9. Remark. All the fields of prime degree I so far known with an integer normal basis generated by a unit have a prime conductor and they are generated by a Gaussian period a. For / = 3, Q C K = Q(C 3 ) and Q C K C Q(C 5 ). For / = 5, QC K C Q(Cn) and Q C K C Q(C 7 i)- For primes equal to or 4/ + 1 fields K of degree /, Q C K C Q(C P ), have an integral normal basis generated by a unit, because a sum of two or four roots of unity is a unit. 147

7 ANTONÍN DVOŘÁK DAVID JEDELSKÝ JURAJ KOSTRA Results for m = 911 and m = 4733, fields K 10 and K 2. L V P Pi , , , , L V P Pi , ,

8 THE FIELDS OF DEGREE SEVEN OVER RATIONALS Results for m = 29 * 4733, fields K 3, K 4 and K b. L V P Pi , , , , ,127, L V P Pi L V P Pi 1 7, lť -Л , , , , , ,

9 ANTONÍN DVOŘÁK DAVID JEDELSKÝ JURAJ KOSTRA Results for m = 29 * 4733, fields K 6, K 7 and K s. 1 V P Pi ,127, í V P Pi , , i V P Pi ,

10 THE FIELDS OF DEGREE SEVEN OVER RATIONALS Results for m = 113 * 911, fields K u, K 12 and K V P Pi ,127, ,127, Í. V P Pi , , ь V P Pi ,

11 ANTONÍN DVOŘÁK DAVID JEDELSKÝ JURAJ KOSTRA Results for m = 113 * 911, fields K 14, K l5 and K 16. L V P Pi L V P Pi L V P Pi , , ,

12 THE FIELDS OF DEGREE SEVEN OVER RATIONALS REFERENCES [1] JAKUBEC, S KOSTRA, J. NEMOGA, K.: On the existence of an integral normal basis generated by a unit in prime extensions of rational numbers, Math. Comp. 56 (1991), [2] JAKUBEC, S KOSTRA, J.: A note on normal bases of ideals, Math. Slovaca 42 (1992), Received July 16, 1997 * IRAFM and Department of Mathematics University of Ostrava Bráfova 7 CZ Ostrava CZECH REPUBLIC dvoraka@osu.cz ** IRAFM University of Ostrava Bráfova 7 CZ Ostrava CZECH REPUBLIC jedelsky@osu.cz *** Department of Mathematics University of Ostrava Bráfova 7 CZ Ostrava CZECH REPUBLIC kostra@osu.cz 153