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On Some Metabelian 3-Groups Realizable and Principalization Gulf Journal of Mathematics Vol 4, Issue 4 (2016) 155-165 ON SOME METABELIAN 3-GROUPS REALIZABLE AND PRINCIPALIZATION AISSA DERHEM1, MOHAMED TALBI2 AND MOHAMMED TALBI3∗ Abstract. Let G be some metabelian 3-group with abelianisation of type (3; 3). In this paper, we prove that G is realizable with some fields k which is the normal closure of a pure cubic field and we apply these results over G to study the capitulation problem of the 3-ideal classes of k. 1. Introduction Let k be an algebraic number field. We denote by Ok, Ek and Cl(k), the ring of integers, the unit group and the ideal class group of k, respectively. For a prime (1) number p, let Clp(k) be the p-class group and kp the Hilbert p-class field of k. (n) (0) (n+1) (n) (1) Further, we define kp , for an integer n > 0, by kp = k and kp = (kp ) . So we have the sequence (1) (n) k ⊆ kp ⊆ ::: ⊆ kp ⊆ ::: that is called the p-class field tower of k. We know that it is finite if and only if there exists a finite p-extension of k whose p-class number is equal to 1. We shall consider a number fields with Cl3(k) is of type (3; 3). The second 3- (2) class group noted by G = Gal k3 =k is metabelian 3-group with abelianisation G/γ2(G) of type (3; 3) where γ2(G) is the derived group of G. By the Galois correspondence and reciprocity law of Artin, it's known that (2) (1) (1) γ2(G) = Gal k3 =k3 ' Cl3(k3 ) is abelian. And (2) (2) (1) (1) G/γ2(G) = Gal k3 =k =Gal k3 =k3 ' Gal k3 =k ' Cl3(k): The four maximal normal subgroups H1;:::;H4 of G are associated with the four unramified cyclic extensions K1;:::; K4 of k of relative degree 3; which are represented by the norm class groups NKi=k(Cl3(Ki)) as subgroups of index 3 in Date: Accepted: Oct 24, 2016. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 11R29,11R11,11R16,11R20; Secondary 20D15. Key words and phrases. Metabelian 3-groups, Groups of coclass 1, pure cubic field, 3-class groups, principalization of 3-classes. 155 156 A. DERHEM, M.TALBI AND M. TALBI the 3-class group Cl3(k) of k and by the Galois correspondence we have Hi = (2) Gal k3 =Ki . p3 2 In the present paper we shall consider k = Q(ζ3; 3q ), the normal closure p3 of the pure cubic field Γ = Q( 3q2), where q is a prime number which verifies q ≡ −1 (mod 9), and ζ3 is the third root of unity. Those fields are of type II in the sense of Isam¨ıli[12]. Isma¨ılialso proved that in this case, there are three type of capitulation (0; 0; 0; 0); (0; 4; 3; 2), (0; 4; 4; 4), and the relative genus field of k over k0, where k0 = Q(ζ3), is one of the four cyclic cubic extension of k, we will noted by K1: We investigate the theory of groups of maximal class and the works [2,4,3,5, (1) 6,7,8, 12, 15, 16, 17, 18], we determine the structure of G1 = Gal (K1)3 =k0 , (4) precisely we show that G1 is of maximal class 4 and G1 = G (0; 1; 0) (with the same notation of Nebelung [18] and Mayer [15]). (2) Further, With the aid of the structure of G1, we prove that also G = Gal k3 =k is of maximal class and there are only one type of capitulation possible which is (2) (0; 0; 0; 0) and the class field tower is finite and terminate at k3 : 2. On decomposition of ideal in number field In this section we develop some results that we need in this paper. A more precision or proof can be found in [1] and [10]. p 3 Let a, b are integers such that ab is square free and ab > 1. Set Γ = Q( ab2). Then an integral basis for Γ is given by: p p3 3 2 ab2 2 2 (1) f1; ab ; b g, if a − b 6≡ 0 (mod 9), p p3 p3 2 3 2 b2±b2 ab2+ ab2 2 2 (2) f1; ab ; 3b g, if a − b ≡ 0 (mod 9). And the discriminant of Γ is given by −27a2b2 if a2 − b2 6≡ 0 (mod 9); d(Γ) = −3a2b2 if a2 − b2 ≡ 0 (mod 9): p 3 Definition 2.1. Let Γ = Q ab2 , be a pure cubic field. We say that Γ is of Kind 1 if a2 − b2 6≡ 0 (mod 9) and of Kind 2 otherwise. In the following proposition we summarize the results concerning the decom- position in pure cubic field. p 3 2 Proposition 2.2. Let Γ = Q ab be a pure cubic field, and OΓ the ring of integer of Γ, and let NΓ=Q be the absolute norm of Γ. 3 (1) If Γ is of Kind 1, then 3OΓ = P , where P is a prime ideal in OΓ. 2 (2) If Γ is of Kind 2, then 3OΓ = P P1, where P and P1 (P 6= P1) are primes ideals in OΓ. (3) If q is a prime number such that q - ab and q 6= 3, then q is unramified in Γ. More precisely, we have: ON SOME METABELIAN 3-GROUPS REALIZABLE 157 (a) If q ≡ −1 (mod 3) then qOΓ = QQ1, with NΓ=Q(Q) = q and NΓ=Q(Q1) = 2 q , where Q and Q1 are primes ideals in OΓ. (b) If q ≡ 1 (mod 3), then: ab2 • if q = 1, then qOΓ = QQ1Q2, with NΓ=Q(Q) = NΓ=Q(Q1) = 3 NΓ=Q(Q2) = q , where Q, Q1 and Q2 are primes ideals in OΓ, ab2 3 • if q 6= 1, then qOΓ = Q, with NΓ=Q(Q) = q where Q is a 3 prime ideal in OΓ.. Theorem 2.3. Let L be a number field whose contains the `-th roots of units, ` a prime number, and θ 2 L such that θ 6= µ` for all µ 2 L. Then p (1) The extension L( ` θ)=L is cyclic of prime degree `. (2) Suppose that prime ideal P of L is above the principal ideal (θ) and define an integer a 2 N by Pa k (θ), then we have: (a) if a = 0, andpP not divides `, then P split completely (resp become prime) in L( ` θ) if and only if θ ≡ ξ` (mod P) is soluble (resp not soluble), p (b) if a is prime to `, then P is totally ramified in L( l θ). (3) Let L denote a prime ideal above ` and define un integer a 2 N by La k (1 − ζ`)p, where ζ` is the primitive `-th root of unity. If θ is prime to `, then L( l θ)=L is unramified at L if and only if θ ≡ ξ` (mod La`). Let p be a prime number such that p ≡ 1 (mod 3), and let k0 the cyclotomic field of third roots of unity. It follows from theorem 2.3 that (for more precision, you can see [13]), (1) If p = π1π2 in k0, π1 and π2 are conjugate, then π1 ≡ π2 ≡ 1 (mod 3Ok0 ). 2 (2) c = c for all c 2 prime to p. π1 π2 Z 3 3 (3) c = c = 1 if and only if c is cubic residue modulo p. π1 π2 3 3 π1 π2 (4) π = π = 1 2 3 1 3 Let us mention that, c 2 Z is cubic residue modulo p means that the congruence X3 ≡ c (mod p) has a solution in Z. Moreover, by applying theorem 2.3 we deduce the decomposition of a prime number q in the normal closure of some cubic pure field. This can be done in straightforward fashion by combining the factorization rules for his cubic pure fieldp and the cyclotomic field contains the third roots of unity. Let k = 3 2 Q(ζ3; ab ) the normal closure of Γ, then the decomposition in k is determined as follows: 3 Proposition 2.4. If q divides ab, then qOk = Q , where Q is a prime ideal in k. And, if q not divides ab, then: (1) Suppose that q ≡ 1 (mod 3), then: (a) if (q) become principal in Γ, then qOk = P1P2, (b) if (q) split completely in Γ, then qOk = P1P2P3P4P5P6. (2) If q ≡ −1 (mod 3), then qOk = P1P2P3. 158 A. DERHEM, M.TALBI AND M. TALBI These results are summarized in thep following: 3 2 A rational prime factors in k = Q(ζ3; ab ) as P1P2:::::Pr, where r is given by: 8 < 3; if q ≡ −1 (mod 3); r = 6; if q ≡ 1 (mod 3) and x3 − ab2 ≡ 0 (mod (q)) is solvable; : 2; Otherwise. 3. The 3-Hilbert field of de genus field of k over k0 Let any finite 3-group G is nilpotent, and γ2(G) his commutator subgroup. Assume that the commutator factor group G/γ2(G) is of type (3; 3), the subgroup 3 G of G generated by the 3-th powers is contained in the commutator group γ2(G), which therefore coincides with the Frattini subgroup j=4 \ 3 Φ(G) = Mj = G γ2(G) = γ2(G); j=1 where Mj; 1 ≤ j ≤ 4 are the maximal normals subgroups of G. According to the basis theorem of Burnside, the group G is generated by two elements.
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