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Additional resources for Algebraic Number Theory: Proceedings of an Instructional Conference Organized by the London Mathematical Society (A Nato Advanced Study Institute W)
The results for non-normal L now follow by embedding L in a normal extension of K. The details are left to the reader. COROLLARY 1. The inertia group To is always soluble. More precisely, 1yx = 0, then r0 is cyclic, and if x = p # 0, then r0 is the extension of a p-group by a cyclic group. If k is finite then the Galois group of a normal extension is soluble. COROLLARY 2. The composite Jield of tamely ramified extensions L and L’ in a separable closure of K is again tamely ramified. The maximal tamely ramified extension K,, of K is the union of all tamely ramified extensions in a separable closure of K.
X is the characteristic of k. The term “subfield” is used as in 7, Theorem 2. TO is always the inertia group de&red there. THEOREM 1. (i) L has a subfield Li such that the subjields L’ of L which are tamely ramified over K are preciseIy the subfieIds of L1. If x = p # 0 then (L : L,) is a power of p. (ii) Suppose that L is normal over K with Galois group I?. Then Ll is normal over K and is thejixedfield of rl = [y E rlv,(xy-x) 2 t+(x)+1 for all x E RL]. IfVL(W = Yz-bmm A. FRGHLICH 30 defines a homomorphism 8,, of rO into kz,, which is independent of the choice of ll and whose kernel is rl.
We may now replace c by c”a’ with (s, e) = 1 and a E K*, and we can thus ensure that Q(C) = 1, and then by (2) (3) E = 8,. If now also L = K(b”‘) then, by Kummer theory, b = c’a’ with a E K*, (r, e) = 1 and 0 < r < e. If vi[(b) = 1 then we must have r = 1, and so the element a is a unit. The final form of our criterion will now follow from 1, Proposition 5. (ii) follows from Kummer theory and by 6, Theorem 1. COROLLARY 1. Let Q(C) = 1. Then K,, is the union of fields K,,,(c”‘), for all e not divisible by x.