On the Ramification Theory of Infinite Algebraic Extensions

Abstract

The ramification theory of the infinite normal algebraic extensions was first treated by J. Herbrand and C. Chevalley [5] in 1933. The Hilbert's ramification groups Q3, of an infinite normal algebraic extension K/k were defined there as the projective limit groups Q~x = proj. lim 58n) of the corresponding ramification groups Q!n) of intermediate finite normal extensions k/Ik; k C ki C : c kn c , K Uw~lkn . It was proved there that the set A of all the ramification groups IQA} are linearly ordered by inclusion and A is metrically complete by a suitable metric topology. It was also proved that all the ramification groups of a normal subfield K'/k (k c K' C K) and of K/k' (k c k' c K) are induced from that of K/k. Since then no remarkable developments and applications of this theory have been made.' Recently T. Tamagawa [11] constructed the conductor theory for Weil's algebroid character and proved a functional equation for the Weil's L-function. He used there essentially the Hilbert's ramification theory of abelian and other related infinite extensions over an algebraic number field of finite degree. Associated with these investigations of Tamagawa, I. Satake [10] also constructed the Hilbert's ramification theory of some types of infinite algebraic extensions, which he called the (H)-extensions, over a field with a discrete or semi-discrete valuation. The purpose of this paper is also to investigate the ramification theory of general infinite separable normal algebraic extensions over a field k with a discrete valuation whose residue class field is perfect. In ?2 we shall give necessary preparations for finite extensions. In ?3 we shall introduce a suitable parameter it for the ramification fields (Theorem 3. A). For this sake we shall define a real canonical set U(K/k) and derive from it two sets S(K/k) and S*(K k;) such that S*(K/k) _ U(K/k) C S(K/k). We shall then classify the ramification fields into two kinds. For the ramification fields of the first kind bKlk(u) we shall introduce a real parameter u e S(K/k), and for that of the second kind UK/k(u) we shall introduce a real parameter ue S*(K/k). We can determine the inclusion and limit relations between these ramification fields (Proposition 3.3, 3.5). The relation between these ramification fields of K/k and of a normal subfield K'/k (or of K/Tk', k c k' c K) can be explicitly given by our parameter u (Theorem 3.B). In ?4 we shall prove a Theorem of Tamagawa [11] (Proposition 4.3) as an application of our theory, and construct some examples of K/k for which the canonical set U(K/k) has the given properties. In ?5, ?6 we shall define the p-exponents of the different [bo(Klk)] and Artin's