18.785F16 Number Theory I Assignments: Problem Set 10
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(a) Prove that for each integer n > 1 there are infinitely many (Z/nZ)-extensions of Q ramified at only one prime. (b) Prove that for each integer n > 2 there are no (Z/nZ)2-extensions of Q ramified at only one prime. Why does this not contradict the fact that (Z/pZ)2-extensions of Qp exists for every p? (c) Given an explicit example of a (Z/2Z)2-extension of Q ramified at only one prime (with a defining polynomial), and prove that this example is unique.
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18.785F16 Number Theory I Lecture 18 Notes: Dirichlet L-functions, Primes in Arithmetic Progressions
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18.785F16 Number Theory I Lecture 26 Notes: Global Class Field Theory, Chebotarev Density
(using the restricted product topology ensures that a 7→ a−1 is continuous, which is not true of the subspace topology). As a topological group, IK is locally compact and Hausdorff. The multiplicative group K× is canonically embedded as a discrete subgroup of IK via the diagonal map x 7→ (x, x, x, . . .), and the idele class group is the quotient CK := IK/K, which is locally compact but not com...
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تاریخ انتشار 2016