Explicit class field theory for rational function fields

Author:
D. R. Hayes

Journal:
Trans. Amer. Math. Soc. **189** (1974), 77-91

MSC:
Primary 12A65; Secondary 12A90

MathSciNet review:
0330106

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Abstract: Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over (the finite field of *q* elements) and the action of the idèle class group via the reciprocity law homomorphism. The theory is closely analogous to the classical theory of cyclotomic extensions of the rational numbers.

**[1]**E. Artin and J. Tate,*Class field theory*, Notes Distributed by the Department of Mathematics, Harvard University, Cambridge, Mass.**[2]**Leonard Carlitz,*A class of polynomials*, Trans. Amer. Math. Soc.**43**(1938), no. 2, 167–182. MR**1501937**, 10.1090/S0002-9947-1938-1501937-X**[3]**Leonard Carlitz,*On certain functions connected with polynomials in a Galois field*, Duke Math. J.**1**(1935), no. 2, 137–168. MR**1545872**, 10.1215/S0012-7094-35-00114-4**[4]***Algebraic number theory*, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR**0215665****[5]**Jonathan Lubin and John Tate,*Formal complex multiplication in local fields*, Ann. of Math. (2)**81**(1965), 380–387. MR**0172878**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0330106-6

Keywords:
Rational function field over a finite field,
explicit class field theory,
cyclotomic extensions

Article copyright:
© Copyright 1974
American Mathematical Society