Pseudofinite fields and reciprocity law (in Ukrainian)

V. Andriuichuk

Let $S$ be nonempty subsets of prime numbers. An extension $L$ of a field $K$ is said to be $S$-extension if all the prime divisors of degree $[L:K]$ are contained in $S$. Let $\varkappa$ be a $S$-pseudofinite field, $K$ be a field of algebraic functions of one variable with the field of constants $\varkappa$, $\mathcal I_K$ be the group of ideles of the field $K$ and $C_K$ be the idel class group, $G_K^{ab}$ be the Galois group of maximal abelian $S$-extension of $K$. It is shown in the paper that the global norm-residue map $\Theta_K:\mathcal I_K\to G_K^{ab}$ induces an isomorphism $ \Theta_{L/K}:C_K/N_{L/K}C_L\to Gal(L/K) $ for every finite abelian $S$-extension of the field $K$.

1. Rim D.S., Whaples G. Global norm-residue map over quasi-finite fields // Nagoya Math. J. 1966. V.27, N.1. P.323–329.

2. Введенский O.H. О локальных полях классов эллиптических кривых // Изв. АН СССР, Сер. Матем. 1973. T.37. C.20–88.

3. Tae Kun Seo, Whaples G. A generalization of global class field theory // Can. J. Math. 1969. V.21. P.609–614.

4. Ax J. The elementary theory of finite fields // Ann. Math. 1968. V. 88, N.2. P.239–271.

5. Serre J.-P. Corps locaux. Paris. Hermann, 1962.

6. Sekiguchi K. Class field theory of $p$-extension over a formal power series field with a $p$-quasifinite coefficient field // Tokyo J. Math. 1983. V.6, N.1. P.167--190.

7. Андрийчук В.И. $S$-квазиконечные поля и поля классов // Международная конференция по алгебре, посвященная памяти А.И.Ширшова. Барнаул, 20--25 августа 1991 г. Тезисы докладов по алгебраической геометрии и применениям алгебры к геометрии, анализу и теоретической физике. С.37.

8. Алгебраическая теория чисел (под. ред. Дж.Касселса и А.Фрелиха).– Москва. Мир, 1969.

9. Lichtenbaum S. Duality theorem for curves over $p$-adic field // Invent. Math. 1969. V.7. P.120--136. Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290000, Ukraine

	Pseudofinite fields and reciprocity law (in Ukrainian)
Author	V. Andriichuk Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290000, Ukraine
Abstract	Let $S$ be nonempty subsets of prime numbers. An extension $L$ of a field $K$ is said to be $S$-extension if all the prime divisors of degree $[L:K]$ are contained in $S$. Let $\varkappa$ be a $S$-pseudofinite field, $K$ be a field of algebraic functions of one variable with the field of constants $\varkappa$, $\mathcal I_K$ be the group of ideles of the field $K$ and $C_K$ be the idel class group, $G_K^{ab}$ be the Galois group of maximal abelian $S$-extension of $K$. It is shown in the paper that the global norm-residue map $\Theta_K:\mathcal I_K\to G_K^{ab}$ induces an isomorphism $ \Theta_{L/K}:C_K/N_{L/K}C_L\to Gal(L/K) $ for every finite abelian $S$-extension of the field $K$.
Keywords	prime numbers, S-extension, field extension, S-pseudofinite field, algebraic function field, ideles, idel class group, Galois group
DOI	doi:10.30970/ms.2.1.14-20
Reference	1. Rim D.S., Whaples G. Global norm-residue map over quasi-finite fields // Nagoya Math. J. 1966. V.27, N.1. P.323–329. 2. Введенский O.H. О локальных полях классов эллиптических кривых // Изв. АН СССР, Сер. Матем. 1973. T.37. C.20–88. 3. Tae Kun Seo, Whaples G. A generalization of global class field theory // Can. J. Math. 1969. V.21. P.609–614. 4. Ax J. The elementary theory of finite fields // Ann. Math. 1968. V. 88, N.2. P.239–271. 5. Serre J.-P. Corps locaux. Paris. Hermann, 1962. 6. Sekiguchi K. Class field theory of $p$-extension over a formal power series field with a $p$-quasifinite coefficient field // Tokyo J. Math. 1983. V.6, N.1. P.167--190. 7. Андрийчук В.И. $S$-квазиконечные поля и поля классов // Международная конференция по алгебре, посвященная памяти А.И.Ширшова. Барнаул, 20--25 августа 1991 г. Тезисы докладов по алгебраической геометрии и применениям алгебры к геометрии, анализу и теоретической физике. С.37. 8. Алгебраическая теория чисел (под. ред. Дж.Касселса и А.Фрелиха).– Москва. Мир, 1969. 9. Lichtenbaum S. Duality theorem for curves over $p$-adic field // Invent. Math. 1969. V.7. P.120--136. Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290000, Ukraine
Pages	14-20
Volume	2
Issue	1
Year	1993
Journal	Matematychni Studii
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