By Serge Lang
This is a moment version of Lang's famous textbook. It covers the entire uncomplicated fabric of classical algebraic quantity concept, giving the scholar the historical past beneficial for the research of additional themes in algebraic quantity idea, equivalent to cyclotomic fields, or modular forms.
"Lang's books are continuously of serious worth for the graduate scholar and the study mathematician. This up to date variation of Algebraic quantity concept isn't any exception."―-MATHEMATICAL REVIEWS
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Additional info for Algebraic Number Theory
Assume that we have extended the absolute value to E in some way. Since E is a finite dimensional vector space over K, it is easy to verify that all extensions of v to E are equivalent, and we shall recall the proof below. Since two of them are positive powers of each other, and since they coincide on K, we conclude that they must be equal. Thus we get: If K is complete under an absolute value, then an extension of this absolute value to a finite extension is uniquely determined. In particular, if E is a [II, U] 33 DEFINITIONS AND COMPLETIONS finite extension of K and (f : E ~ (fE is an isomorphism of E over K, then I(fa I = lai for every a E E.
24 [1, §7] ALGEBRAIC INTEGERS If S is the complement of p in A, then multiplying the above factorization by S gives us the factorization of S-lp in S-lB. ßi remain distinct. ßilp). If we assume that A is a local ring, then p = (11") is principal (Proposition 15). ßi in Band let Bi = gilB = B~i. ßi is principal, generated by an element pB i 7ri, and we have = 1I"B i = (1I"ii). ß above p in B. Prove this as an exercise. Denote by I(A) the group of fractional ideals of a Dedekind ring A. Let K, L, B be as above.
Let LI be the integral closure of A in the algebraic closure K of K. Let cp be a homomorphism of LI such that its restriction to BE has the maximal ideal ~E as kernel. Then the map BE 1-+ B";; induces a bijection between unramified extensions E of K and separable extensions of A"'. Proof. We have shown in Proposition 7 that every finite separable extension of A'" is obtainable as an image B'l; for some finite extension E 50 COMPLETIONS [II, §4] of K, unramified over K. We now must prove the uniqueness.