8410Chapter8

8410Chapter8 - RAY CLASS GROUPS AND RAY CLASS FIELDS: FIRST...

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Unformatted text preview: RAY CLASS GROUPS AND RAY CLASS FIELDS: FIRST CLASSICALLY, THEN ADELICALLY PETE L. CLARK Contents 8. Intro 1 8.1. Moduli and ray class fields 1 8.2. The Hilbert Class Field 5 8.3. Remarks on real places, narrow class groups, et cetera 6 8.4. A word about class field towers 7 8.5. Class field theory over Q 8 8.6. Idelic interpretation 10 References 12 8. Intro We wish to explain our earlier claim that the idele class group C ( K ) is, as a topolog- ical group, a “target group” for global class field theory: i.e., its profinite completion is canonically isomorphic to the Galois group Gal( K ab /K ). Note first that the statement that C ( K ) is such a target group is equivalent to the statement that C 1 ( K ) is such a target group: namely, we have in the number field case a short exact sequence 1 → C 1 ( K ) → C ( K ) → R > → 1 which shows that the inclusion C 1 ( K ) , → C ( K ) induces an isomorphism on the profinite completions. Exercise 8.1: What happens in the function field case? It may therefore be worth mentioning that the object C ( K ) is for many purposes “more fundamental”, whereas the subgroup C 1 ( K ) was – by virtue of being com- pact – technically more convenient to establish the two finiteness theorems of the previous section. 8.1. Moduli and ray class fields. N.B.: On a first reading, I suggest that the reader make her task a little easier by: (i) restricting to the number field case, and (ii) ignoring – or mostly ignoring – the infinite part of the modulus. 1 2 PETE L. CLARK What we can do is define a certain family { K ( m ) } of abelian extensions which are parameterized solely by some arithmetic data m from K (you are not yet sup- posed to know what this means; don’t worry). These field K ( m ) are called ray class fields of K . It is too much to hope for that every finite abelian extension of K is a ray class field, but what turns out to be true is that every abelian extension L is contained in some ray class field – in fact, in infinitely many ray class fields, but there will be a unique smallest ray class field containing L . The Galois the- ory of subextensions of abelian extensions behaves beautifully – in particular every subextension is Galois – so that if we know all the ray class fields, we have a good chance at understanding all the finite abelian extensions. Let me now describe the objects m by which ray class fields are parameterized. Recall that if K/ Q is a number field of degree d , say given as Q [ t ] / ( P ( t )), then the embeddings of K into R correspond precisely to the real roots of P ( t ): in par- ticular there is somewhere between 0 and [ K : Q ] such embeddings. Let us label these embeddings ∞ 1 , ∞ 2 ,..., ∞ r . We call such embeddings “real places.” The function field case is simpler: there are no real embeddings to worry about....
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This note was uploaded on 10/26/2011 for the course MATH 8410 taught by Professor Staff during the Fall '11 term at UGA.

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8410Chapter8 - RAY CLASS GROUPS AND RAY CLASS FIELDS: FIRST...

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