Gauss sums for \({\mathbb F}_ q[T]\). (English) Zbl 0629.12014
‘Gauss and Jacobi sums’, taking values in function fields of one variable over a finite field, are defined using the Carlitz-Hayes-Drinfeld cyclotomic theory and the usual cyclotomic theory for \({\mathbb{F}}_ q[T]\). Analogues of Stickelberger’s theorem, the Hasse-Davenport theorem, Weil’s theorem on ‘Jacobi sums as Hecke characters’ and the Gross-Koblitz theorem are proved.
MSC:
| 11R58 | Arithmetic theory of algebraic function fields |
| 11R18 | Cyclotomic extensions |
| 11T23 | Exponential sums |
| 11T22 | Cyclotomy |
Keywords:
Gauss sums; function fields over finite field; Carlitz-Hayes-Drinfeld cyclotomic theory; Stickelberger’s theorem; Hasse-Davenport theorem; Weil’s theorem; Jacobi sums; Hecke characters; Gross-Koblitz theoremReferences:
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| [2] | [Ca 2] Carlitz, L.: A class of polynomials. Trans. Am. Math. Soc.43, 167-182 (1938) · JFM 64.0093.01 · doi:10.1090/S0002-9947-1938-1501937-X |
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