The Chebotarev Density Theorem
The major connection between the theory of finite fields and the arithmetic of number fields and function fields is the Chebotarev density theorem. Explicit decision procedures and transfer principles of Chapters 20 and 31 depend on the theorem or some analogs. In the function field case our proof, using the Riemann hypothesis for curves, is complete and elementary. In particular, we make no use of the theory of analytic functions. The number field case, however, uses an asymptotic formula for the number of ideals in an ideal class, and only simple properties of analytic functions. In particular, we do not use Artin’s reciprocity law (or any equivalent formulation of class field theory). This proof is close to Chebotarev’s original field crossing argument, which gave a proof of a piece of Artin’s reciprocity law for cyclotomic extensions.
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