Algebra 2

Introductory course on field theory, with Galois correspondence theorem as the main result.

Master the basic tools of commutative algebra and introduce a typical correspondence theorem, which is very useful in mathematics.

A Bachelor's degree in Mathematics.

Recommended prerequisites: the content of the two L3 courses "Groups and Rings 1" and "Groups and Rings 2" from the Bachelor's degree in Mathematics at the University of Montpellier.

1. Revisions on rings, fields; prime subfields, characteristic of a field, Frobenius morphism, factorization, and Eisenstein's criterion.
2. Field extensions: degree formula, algebraic extensions, algebraically closed fields, algebraic closures, splitting fields, decomposition fields, extensions of field morphisms.
3. The Galois group; invariant subfields, Artin's theorem.
4. Finite fields: Galois group, subfields, Galois correspondence.
5. Normal extensions, those that are finished are decomposing bodies.
6. Polynomials and separable extensions: definitions, composition of separable extensions, perfect fields (characterizations), primitive element theorem.
7. Galois extensions: definition(s), conjugate elements. Galois correspondence; examples and applications.
8. Solving polynomial equations: Galois group of a polynomial, action on roots, Galois theorem on solvability by radicals in characteristic zero.

Hourly volumes:

            CM: 9 p.m.

            TD: 9:00 p.m.

            TP: 0

            Land: 0