Class Field Theory by Jürgen Neukirch (auth.)

By Jürgen Neukirch (auth.)

Class box thought, that is so instantly compelling in its major assertions, has, ever when you consider that its invention, suffered from the truth that its proofs have required a sophisticated and, by way of comparability with the consequences, fairly imper­ spicuous procedure of arguments that have tended to leap round in every single place. My prior presentation of the idea [41] has reinforced me within the trust hugely intricate mechanism, corresponding to, for instance, cohomol­ ogy, will not be sufficient for a number-theoretical legislations admitting a truly direct formula, and that the reality of this kind of legislation needs to be at risk of a much more instant perception. i used to be made up our minds to write down the current, new account of sophistication box conception via the invention that, actually, either the neighborhood and the worldwide reciprocity legislation should be subsumed below a merely staff­ theoretical precept, admitting a completely ordinary description. This de­ scription makes attainable a brand new starting place for the full concept. The fast improve to the most theorems of sophistication box idea which ends up from this strategy has made it attainable to incorporate during this quantity an important outcomes and embellishments, and extra comparable theories, with the excep­ tion of the cohomology model which i've got this time excluded. This is still an important version, wealthy in software, yet its central effects may be at once bought from the fabric handled here.

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1) Theorem. The map Lf-+%L =NL1KL* is a 1 -i-correspondence between the finite abelian extensions L of a local field K and the subgroups % of finite index in K*. Moreover Proof By Chap. 2) we have only to show that the subgroups % of K*, which are open in the norm topology, are precisely the subgroups of finite index in K*. 1). Conversely, let (K*: %) = m be finite. Then %;2 K*m and it suffices to prove that K*m contains a norm group. For this we use Kummer theory (see Chap. I, § 5). We may assume that K* contains the group 11m of m-th roots of unity.

Pp-l This shows that xv/v! I_e_ then exp maps p~ into U~). (1 + y))=log(1 +x)+log(1 + y). Namely, these are formal power series identities, and all senes converge. 0 This proves the theorem. We now prove that the class field axiom holds true for p-adic number fields, where we can use the p-adic logarithm. For local fields of finite characteristic the proof needs another method for which we refer to [9].

Vi) If (a/) = 1 for all a norm of the extension bEK*, then K(Vb)IK. aEK*". Proof. (i) and (ii) are clear by the definition, and (vi) expresses the nondegeneracy of the Hilbert symbol. 1) which shows that (a/) = 1 iff (a, K(Vb)IK)= 1. =O, and if ( is a primitive n-th /I-I Let d be the greatest divisor of n such that l = b has a solution in K, and let n=d·m. The extension K(fJ)IK is cyclic of degree m, and the conjugates of x - (i fJ are the elements x - (j fJ with j == i mod d. Therefore x" - b = d-l TI i=O NK(Plldx - (i fJ), Chapter III.

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