By Serge Lang

ISBN-10: 0387942254

ISBN-13: 9780387942254

This is a moment version of Lang's recognized textbook. It covers the entire uncomplicated fabric of classical algebraic quantity conception, giving the coed the heritage beneficial for the learn of extra issues in algebraic quantity concept, resembling cyclotomic fields, or modular forms.

"Lang's books are regularly of significant price for the graduate scholar and the learn mathematician. This up-to-date version of Algebraic quantity thought is not any exception."―-MATHEMATICAL REVIEWS

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**Example text**

From now on, by fractional ideal we shall mean non-zero fractional ideal, unless otherwise specified. Let A be a Dedekind ring and a a fractional ideal. W e have a factorization with integers rp all but a finite number of which are O. We say that rp is the order of a at ~· If rp > O, we say that a has a zero at ~· If rp < O, we say that it has a pole at ~· Let a be a non-zero element of the quotient field of A. Then we can form the fractional ideal (a) = Aa and we apply the above notions of order, zero, and pole to a.

If A is a Dedekind ring, and p a prime ideal of A, then Ap is a discrete valuation [1, §7] DISCRETE VALUATION RINGS 23 ring, since it is equal to s- 1 A (S = complement of ll in A) (cf. Proposition 16). Since every ideal of a discrete valuation ring is principal, it must be some power of the maximal ideal. In proving theorems about Dedekind rings, it is frequently useful to localize with respect to one prime ideal, in which case one obtains a discrete valuation ring. For instance we have the following proposition.

Let A= Aj be the projection on the j-th coefficient. (M) =/:- {0}. After multiplying A. (M) cA. (M) = a1. Then a1 is an ideal, and we have a surjective homomorphism M -+ a1 giving rise to an exact sequence By Proposition 26, a1 is projective, so M ~ a1 $ M'. We leave it to the reader to verify that the rank of M' is r-1, so the proof of the first statement is concluded by induction. If one takes for A. a functional such that A(M) is maxima! among ali possible ideals obtained as above, then the inductive sequence of ideals a17 ••• , ar which one obtains satisfies aii ai+l for ali i.

### Algebraic Number Theory by Serge Lang

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