On some class number relations for Galois extensions

Abstract

Let k be an algebraic number field of finite degree over Q , the field of rationals, and K be an extension of finite degree over k . By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k ) for the extension K/k . When k = Q and K is quadratic over Q , the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and t K = the number of distinct prime factors of the discriminant of K , may be considered as an equality between E(K/Q ) and other arithmetical invariants of K .

Locations

• Nagoya Mathematical Journal - View - PDF
• Project Euclid (Cornell University) - View - PDF