Sur le rang du 2-groupe de classes de 𝑄({√{𝑚}},{√{𝑑}}) où 𝑚=2 ou un premier 𝑝≡1(𝑚𝑜𝑑4)

Abstract

SUR LE RANG DU 2-GROUPE DE CLASSES DEABDELMALEK AZIZI AND ALI MOUHIB Abstract. On the rank of the 2-class group of Q(Let d be a square-free positive integer and p be a prime such that p ≡ 1 (mod 4). We, where m = 2 or m = p. In this paper, we determine the rank of the 2-class group of K., un corps biquadratique où m = 2 ou bien un premier p ≡ 1 (mod 4) et détant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du 2-groupe de classes de K.

Real quadratic number fields with metacyclic Hilbert $2$-class field tower

“…Since G/N is abelian, G ′ ⊂ N and then G ′ is cyclic. We deduce that K (∞) /K (1) is abelian unramified. So K (∞) ⊂ K (2) and then K (∞) = K (2) .…”

“…Let K be a number field and p a prime integer. (1) is the largest abelian extension of K contained in K (∞) . We deduce that Gal(K (∞) /K (1) ) ∼ = G ′ .…”

“…(1) is the largest abelian extension of K contained in K (∞) . We deduce that Gal(K (∞) /K (1) ) ∼ = G ′ . Let N be a normal cyclic subgroup of G such that G/N is cyclic.…”

“…Let K be a number field and C K be the class group of K. The maximal unramified abelian extension of K denoted by K (1) is called the Hilbert class field of K. We recall that by the Artin reciprocity law we have Gal(K (1) /K) ≃ C K . For a nonnegative integer n, let K (n) be defined inductively as K (0) = K and K (n+1) = (K (n) ) (1) ; then K ⊂ K (1) ⊂ K (2) ⊂ . .…”

“…p , the maximal unramified abelian p-extension of K, is called the Hilbert p-class field of K. We recall that by the class field theory we have Gal(K p ; then K ⊂ K (1) p ⊂ K (2) p ⊂ . .…”