Pro-pGalois Groups of Algebraic Extensions of Q

✍ Scribed by Ido Efrat

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 394 KB
• Volume: 64
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1997.2091

✦ Synopsis

For a prime number p we characterize the finitely generated maximal pro-p Galois groups of algebraic extensions of Q. This generalizes a characterization by Jensen and Prestel of the maximal abelian quotients of these Galois groups. As an application we show that the Witt rings of the algebraic extensions of Q with finitely many square classes have elementary type. 1997 Academic Press A presentation of the groups of type (iii) by pro-p generators and one relation (depending on E) has been given by Demus kin [D1, D2], Serre [Se1], and Labute [L1].

The theorem above generalizes results of Jensen and Prestel [JP1, Theorems 3.2 and 3.2$], who characterize the finitely generated maximal pro-p abelian Galois groups G K ( p, ab) of algebraic extensions K of Q. Note that G K ( p) is finitely generated if and only if G K ( p, ab) is finitely generated (see Corollary 4.6).