Class Field Theory
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Let k be a finite field with q = pn elements and let V be an algebraic variety defined over k (or, as one also says, a k-variety). Suppose that V is defined by charts Ui (isomorphic to affine k-varieties) and changes of coordinates uij (with coefficients in k). If x = (x1, …, xr) is a point of an affine space, we write Fx, or xq, for the point with coordinates (xq1, …, xqr). The map x → Fx commutes with polynomial maps with coefficients in k. In particular, it maps each of the Ui, into itself and commutes with the Uij; therefore by “glueing” it operates on V. The image of a point x ∈ V will again be denoted Fx or xq.
KeywordsExact Sequence Homogeneous Space Algebraic Group Galois Group Finite Index
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