By Serge Lang
This is a moment variation of Lang's recognized textbook. It covers all the easy fabric of classical algebraic quantity concept, giving the coed the history beneficial for the research of additional themes in algebraic quantity concept, similar to cyclotomic fields, or modular forms.
"Lang's books are regularly of significant price for the graduate scholar and the study mathematician. This up to date variation of Algebraic quantity conception is not any exception."―-MATHEMATICAL REVIEWS
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Since the absolute values are assumed tobe non-trivial, there exists y E K such that JyJ 1 > 1. Let a= JyJ 1and let b = IYI2· Let x E K, x =7= O. Say JxJ ~ 1. Then JxJ1 = JyJ~ for some a ~ O. If m, n are integers > O such that m/n > a, we have JxJ1 < whence Jxn/ymJ 1 < JyJ'f 1n, 1, and thus Jxn/ymJ2 < Similarly if m, n are integers such that m/n JxJ2 > Hence JxJ 2 = JyJ~. 1, so that < a, then JyJ~fn. From this it follows immediately that JxJ1 = JxJ~, where X = (log a)/(log b), thus proving our assertion.
Um) -_ (U: . ) _ ( . 1). 1 - Ur. Ur+s)(Km But (Ur: Ur+•) = (Np)• and our assertion follows. 48 COMPLETIONS [II, §4] Corollary. lf K contains the m-th roots of unity, then and §4. Unramified extensions lVe continue to assume that K is complete under a discrete valuation, with ring A and maxima[ ideal\). If E is a fin~te extension of K and B the integral closure of A in E, then there is a unique prime ideal '13 of B lying above \), and B is a discrete valuation ring. If e is the ramification index and f the residue class degree, then ef = [E :K].
E. completion of a number field under a )l-adic valuation). Let E be an unramified extension of K. Then every unit of K is a norm of a unit in E. Proof. Let u bea unit in K. We identify the Galois group of E over K with the Galois group of the residue class field extension. It is a simple consequence of Hilbert's Theorem 90 (or anything else you can think of) that both the trace and norm from a finite extension of a finite field are surjective. Hence there exists a unit a 0 in E such that u Then for some c1 E A.
Algebraic Number Theory by Serge Lang