Construction of somep-extensions of the rational number field

Abstract

In this paper, we constructp-extensionsK a ,a(modp r ), of degreep 3r,p≠2, r>0, of the field ℚ of rational numbers with ramification pointsp andq. The Galois groupG(K a )/ℚ of the extensionK a /ℚ,a(modp r ), is defined by the generators and relations $$\sigma ^{p^r } = 1, \tau ^{p^{2r - n} } = 1, \tau ^{p^r } [\tau ,\sigma ]^a = 1, G^{(2)} (K_a /\mathbb{Q}) = \{ 1\}$$ , where the numbern is such thatp n |a andp n+1βa. The form of the relation between two generators of the Galois groupG p (p, q) of the maximalp-extension with two ramification pointsp andq modulo the third term of the descending central series of this group depends on the character of the decomposition of the numberq in the fieldsK a ,a(modp r ).