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Unformatted text preview: 8430 HANDOUT 5: CHEBOTAREV DENSITY; GLOBAL CLASS FIELD THEORY PETE L. CLARK Remark: This handout presents some of the most important results of algebraic number theory. Although our intended application is to the case of K an imagi nary quadratic field, L/K a certain finite abelian extension, R = O K and S = O L , whenever it was not obviously inconvenient I have presented the results in more generality. Some of my motivations for doing this are as follows: first, in contrast to the intended “basic graduate level” audience of Part II of Cox’s book, this course is a topics course which – while, I hope, being mostly accessible to students with a background in basic graduate algebra – is also intended to be useful for students who are now doing or contemplate going on to do thesis work in algebraic number theory and/or arithmetic geometry. For such students – i.e., for at least three out of five – the extra generality I add over Cox’s treatment will almost certainly come in handy later in your career. Moreover, Dino has recently taught two topics courses which develop things in a similar level of generality (and, in fact, with even more loving attention to inseparable field extensions), so it would be a disservice to stu dents who took either or both of his courses not to give the natural continuations of some of the topics from Dino’s courses. My recommendation to you is to pay especially close attention to the Frobenius elements and Chebotarev Density – if you really understand the welldefinedness of Frobenius elements for unramified primes up to conjugacy, you will be well equipped to appreciate the remarkable simplicity of Chebotarev’s theorem: it just says that the probability that a conjugacy class C in Gal( L/K ) is the Frobenius conjugacy class of a randomly chosen prime ideal p is exactly what equidistribution would predict: # C #Gal( L/K ) . What could be simpler? In contrast, no one has ever accused class field theory of being simple. I think it is fair to say that the material presented in this handout will be best understood and appreciated by patiently parsing the complicated statements and then waiting to see how it will be applied (which is now coming up very soon). You are entitled to be a little confused! 1. Decomposition of ideals in separable extensions The running hypotheses for this section are as follows: R is a Dedekind domain with quotient field K , L/K is a separable field extension of degree n , and S is the integral closure of R in L . Recall that S is a Dedekind domain and (by virtue of the separability) is finitely generated as an Rmodule. If p is a prime of R , then because the extension is integral it follows that p S is a proper ideal of S . Therefore 1 2 PETE L. CLARK it factors, say as p S = g Y i =1 P e i i ....
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 Fall '11
 Clark
 Algebra, Geometry, Number Theory, Algebraic number theory, Class field, pete l. clark, abelian extension, Chebotarev Density

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