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8410Chapter7

# 8410Chapter7 - THE IDELIC APPROACH TO CLASS GROUPS AND UNIT...

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THE IDELIC APPROACH TO CLASS GROUPS AND UNIT GROUPS PETE L. CLARK Contents 7.1. Rings of S -integers 1 7.2. Finiteness of the S -class groups 1 7.3. Structure of the S -unit groups 5 7.1. Rings of S -integers. Notation: Despite some misgivings, we introduce the following notation. For a global field K , we let Σ K denote the set of all places of K , i.e., equivalence classes of nontrivial norms on K . (Recall that at this point we have associated a canon- ical normalized Artin absolute value to each place in K .) Further we write Σ NA K for the subset of non-Archimeden places of K , and let Σ Arch K denote the subset of Archimedean places of K , which is nonempty iff K is a number field. Now let S Σ K be a finite set such that S Σ Arch K . We define the ring R S of S-integers of K to be the set of x K such that v ( x ) 0 for all v S \ Σ Arch K . Exercise 7.1: a) If K is a function field and S = , show that R S is equal to the maximal finite subfield of K (or, in other words, the “algebraic closure of F p in K ”.) b) Show that if S = , then R S is a Dedekind domain that is not a field. Exercise 7.2: Suppose that S T are finite subsets of Σ K with Σ Arch K S . a) Show that R S R T . b)* Show that R T is a localization of R S . (The special case that R S is a PID is straightforward. In order to do the general case, it may be necessary to use the finiteness of the ideal class group of R S , coming up shortly.) 7.2. Finiteness of the S -class groups. Because R S is a Dedekind domain, it has an interesting invariant attached to it: its ideal class group Cl( R S ). This is the free abelian group generated by the nonzero prime ideals, which we may view as being in correspondence with the non- Archimedean places v of K . Inside Frac( R S ) we have the subgroup of principal fractional ideals ( f ) for f K × . Explicitly, we view ( f ) as an element of v Z [ v ] by sending f to the sequence { v ( f ) } , which has all but finitely many elements 0. Let 1

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2 PETE L. CLARK us write this subgroup as Prin( K ). By definition, the ideal class group Cl( R S ) of R S is the quotient Frac( R S ) / Prin( R S ). To summarize, we have exact sequences 1 R × S K × Prin( R S ) 1 , 1 Prin( R S ) Frac( R S ) Cl( R S ) 1 . Note that this discussion makes perfect sense for an arbitrary Dedekind domain R . Indeed, a celebrated theorem of L. Claborn asserts that for any abelian group A whatsoever, there exists a Dedekind domain R such that Cl( R ) = A . However, in the arithmetic case, we have the following important result, a generalization of the second of the three fundamental finiteness theorems in classical algebraic number theory. Theorem 1. Let K be a global field and S Σ K a finite set containing all the Archimedean places. Then the ideal class group Cl( R S ) = Frac( K ) / Prin( K ) is a finite abelian group.
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8410Chapter7 - THE IDELIC APPROACH TO CLASS GROUPS AND UNIT...

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