On the rank of the $$2$$ 2 -class group of $$\mathbb {Q}(\sqrt{p}, \sqrt{q},\sq
- PDF / 211,667 Bytes
- 8 Pages / 439.37 x 666.142 pts Page_size
- 87 Downloads / 246 Views
√ √ √ On the rank of the 2-class group of Q( p, q, −1) Abdelmalek Azizi · Mohammed Taous · Abdelkader Zekhnini
© Akadémiai Kiadó, Budapest, Hungary 2014
√ √ (2) Abstract Let d be a square-free integer, k = Q( d, i) and i = −1. Let k1 be the (2) (2) (2) Hilbert 2-class field of k, k2 be the Hilbert 2-class field of k1 and G = Gal(k2 /k) be the (2) Galois group of k2 /k. We give necessary and sufficient conditions to have G metacyclic in the case where d = pq, with p and q primes such that p ≡ 1 (mod 8) and q ≡ 5 (mod 8) or p ≡ 1 (mod 8) and q ≡ 3 (mod 4). Keywords
2-class groups · Hilbert class fields · 2-metacyclic groups
Mathematics Subject Classification
11R11 · 11R29 · 11R32 · 11R37
1 Introduction (1)
Let k be an algebraic number field and let Cl2 (k) denote its 2-class group. Denote by k2 the (2) (2) Hilbert 2-class field of k and by k2 its second Hilbert 2-class field. Put G = Gal(k2 /k) and G its derived group. It is well known that C/G Cl2 (k). An important problem in Number Theory is to determine the structure of G, since the knowledge of G, its structure and its generators, among other things, solves capitulation problems, helps decide whether class field towers are finite or not and the structures of the 2-class groups of the unramified (1) extensions of k within k2 . Often, the rank of G is sufficient to determine the structure of G. In this paper, we give an example of this situation.
A. Azizi · A. Zekhnini Département de Mathématiques, Faculté des Sciences, Université Mohammed 1, Oujda, Morocco e-mail: [email protected] A. Zekhnini e-mail: [email protected] M. Taous (B) Département de Mathématiques, Faculté des Sciences et Techniques, Université Moulay Ismail, Errachidia, Morocco e-mail: [email protected]
123
A. Azizi et al.
√ Let k = Q( pq, √ i), where p and q are two different primes; then the genus field of k is √ √ k∗ = Q( p, q, −1). According to [7] r0 , the rank of the 2-class group of k, is at most (2) equal to 3. Moreover, r0 = 3 if and only if p ≡ q ≡ 1 (mod 8). Let G = Gal(k2 /k) be (2) (i+1) (i) is the Hilbert 2-class field of k2 , with i = 0 or 1 the Galois group of k2 /k, where k2 (0) and k2 = k. The Artin Reciprocity implies that r0 = d(G), where d(G) is the rank of G. In [7], the first and the second authors have shown that if q = 2, then G is metacyclic non-abelian if and only if p = x 2 + 32y 2 and x ≡ ±1 (mod 8). In this paper, we prove that if p and q are odd different primes, then r , the rank of the 2-class group of k∗ , helps to know in which case G is metacyclic. 2 The rank of the 2-class group of k∗ In what follows, we adopt the following notations: If p ≡ 1 (mod 8) is a prime, then
p−1
2 p 4
will denote the rational biquadratic symbol which is equal to 1 or −1, according as 2 4 ≡ ±1 p−1 (mod p). Moreover, the symbol 2p 4 is equal to (−1) 8 . Let k be a number field and l be x, y a prime; then lk will denote a prime ideal of k above l. We denote, also, by resp. lk x the Hilbert symbol (resp. the quadratic residue symbol) for the pr
Data Loading...
