The elementary theory of finite fields

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We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial. Then we shall do a bit of commutative algebra finite algebras over a field, base change via tensor product and apply this to study the notion of separability in some detail. After that we shall discuss Galois extensions and Galois correspondence and give many examples cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc.

We shall address the question of solvability of equations by radicals Abel theorem.

We shall also try to explain the relation to representations and to topological coverings. Finally, we shall briefly discuss extensions of rings integral elemets, norms, traces, etc. Some knowledge of commutative algebra prime and maximal ideals — first few pages of any book in commutative algebra is welcome.

For exercises we also shall need some elementary facts about groups and their actions on sets, groups of permutations and, marginally, the statement of Sylow's theorems. Please help to improve this article by introducing more precise citations. December Learn how and when to remove this template message. Categories : Model theory Field theory.

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Each Element in a Finite Field is the Sum of Two Squares

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Each Element in a Finite Field is the Sum of Two Squares | Problems in Mathematics

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