Get A Brief Introduction to Classical and Adelic Algebraic PDF

By Stein W.

Show description

Read Online or Download A Brief Introduction to Classical and Adelic Algebraic Number Theory PDF

Similar algebra books

Get Poxvirus IFNα/β Receptor Homologs PDF

The invention of extracellular poxvirus inhibitors of sort I interferons in 1995 used to be made via direct inhibition reports, instead of by means of series homology research. actually, the vaccinia virus (strain Western Reserve) prototype of this relations, B18R, is extra heavily concerning individuals of the Ig superfamily than to the mobile variety I interferon receptors, a minimum of when it comes to total similarity ratings.

Extra resources for A Brief Introduction to Classical and Adelic Algebraic Number Theory

Sample text

R := PolynomialRing(RationalField()); > K := NumberField(x^3-2); > O := Order([2*a]); > O; Transformation of Order over Equation Order with defining polynomial x^3 - 2 over ZZ Transformation Matrix: [1 0 0] [0 2 0] [0 0 4] > OK := MaximalOrder(K); > Index(OK,O); 8 > Discriminant(O); -6912 > Discriminant(OK); -108 > 6912/108; 64 // perfect square... 2. 4 45 Ideals > R := PolynomialRing(RationalField()); > K := NumberField(x^3-2); > O := Order([2*a]); > O; Transformation of Order over Equation Order with defining polynomial x^3 - 2 over ZZ Transformation Matrix: [1 0 0] [0 2 0] [0 0 4] > OK := MaximalOrder(K); > Index(OK,O); 8 > Discriminant(O); -6912 > Discriminant(OK); -108 > 6912/108; 64 // perfect square...

As an abelian group OK is free of rank equal to the degree [K : Q] of K, and I is of finite index in OK , so I can be generated as an abelian group, hence as an ideal, by [K : Q] generators. The following proposition asserts something much better, namely that I can be generated as an ideal in OK by at most two elements. 7. Suppose I is a fractional ideal in the ring OK of integers of a number field. Then there exist a, b ∈ K such that I = (a, b). Proof. If I = (0), then I is generated by 1 element and we are done.

Every ideal is contained in a maximal ideal, so I is contained in a nonzero prime ideal p. 6 we can cancel I from both sides of this equation to see that p−1 = OK , a contradiction. Thus I is strictly contained in Ip−1 , so by our maximality assumption on I there are maximal ideals p1 , . . , pn such that Ip−1 = p1 · · · pn . Then I = p · p1 · · · pn , a contradiction. Thus every ideal can be written as a product of primes. Suppose p1 · · · pn = q1 · · · qm . If no qi is contained in p1 , then for each i there is an ai ∈ qi such that ai ∈ p1 .

Download PDF sample

A Brief Introduction to Classical and Adelic Algebraic Number Theory by Stein W.


by James
4.2

Rated 4.81 of 5 – based on 22 votes

Categories: Algebra