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Unformatted text preview: Math 676. Global extensions approximating local extensions 1. Motivation A very useful tool in number theory is the ability to construct global extensions with specified local behavior. This problem can arise in several different forms. For example, if F is a global field and v 1 ,...,v n is a finite set of distinct places of F equipped with finite separable extensions K i /F v i , and if S is an auxiliary finite set of non-archimedean places of F , does there exist a finite separable extension F /F unramified at S such that the completion of F at some place v i over v i is F v i-isomorphic to K i ? Moreover, if all extensions K i /F v i have degree d then can we arrange that [ F : F ] has degree d ? If so, are there infinitely many such F (up to F-isomorphism)? An alternative problem is the following: if the extensions K i /F v i are Galois (and so have solvable Galois group, which is obvious when v i is archimedean and which we saw in class in the non-archimedean case via the cyclicity of the Galois theory of the finite fields), then can we arrange for F /F as above to also be Galois with solvable Galois group? If the extensions K i /F v i are abelian (resp. cyclic) then can we arrange for F /F is have abelian (resp. cyclic) Galois group? A typical example of much importance in practice is this: construct a totally real solvable extension L/ Q that is unramified over a specified finite set of primes S of Q and induces a specified extension of Q p for p in another disjoint finite set of primes. Here we are prescribing the local behavior of L at a finite set of primes p and also at the infinite place while insisting that L also be unramified at the auxiliary finite set S . In this handout we provide an affirmative answer to all such construction problems, though we must note at the outset that whereas the answer to the first problem (construction of F /F with prescribed local behavior but no Galois conditions) will only require the weak approximation theorem (as in § 9 of the handout on absolute values), the solution of the problems involving Galois groups will reduce to the abelian case whose solution requires class field theory (and so we will give suitable references in the book of Artin–Tate). To indicate the subtle nature of such construction problems, consider the cyclic case of the preceding questions: if K i /F v i is cyclic with degree n i for all i , can we find a cyclic extension F /F inducing K i /F v i for each i ? If such a construction can be done, then the cyclic group Gal( K i /F v i ) is the decomposition group at v i in the cyclic group Gal( F /F ), and so [ F : F ] must be divisible by the least common multiple n of the n i ’s. Can one always construct such a cyclic extensions F /F with this minimal possible degree? It turns out that this is a subtle problem and in very special circumstances it has a negative answer, and then the best that can be done is [ F : F ] = 2 n . These special circumstances are exhaustively studied in Chapter....
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- Fall '11
- Number Theory, Galois theory, Galois, finite separable extension