On Galois Groups of Local Fields
Kenkichi Iwasawa · Transactions of the American Mathematical Society · 1955
IWASAWALet p be a prime number, Qp the field of p-adic numbers, and fl an algebraic closure of Qp.In the present paper, we take a finite extension k of Qv in ft as the ground field and study the structure of the Galois group G(Q/k) of the extension Q./k.Let V he the ramification field of Q/k, i.e. the composite of all finite tamely ramified extensions of k in Q, and let G(&/ V) and G( V/k) denote the Galois groups of the extensions Q/F and V/k respectively.We shall first determine the structure of the groups Gsplits.Our main result is, then, to describe explicitly the effect of inner automorphisms of G(Q./k) on the factor group of G(Q/V) modulo its commutator subgroup, i.e., on the Galois group G(V'/V) of the maximal abelian extension V of V in ft.This is, of course, not sufficient to determine the structure of the group G(Sl/k) completely; to do that, we still have to find the effect of inner automorphisms of G(ti/k) on the normal subgroup G(Q/V) itself.However, it gives us some insight into the structure of G(il/k); and we hope it will help somehow, in the future, in the study of the group G(ft/&) as well as in that of the Galois groups of algebraic number fields.An outline of the paper is as follows: in §1 we prove some group-theoretical lemmas which will be used later.In §2 we study the behavior of the Galois group of a certain type of finite tamely ramified Galois extension E oi k acting on the multiplicative group of E. Using those results, we then prove in §3 the properties of G(Sl/k) as mentioned above(1).