Algebra 2

An introductory course on field theory, with Galois's correspondence theorem as the main result.

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

A bachelor's degree program in mathematics.

Recommended prerequisites: the material covered in the two third-year courses, “Groups and Rings 1” and “Groups and Rings 2,” in the Bachelor of Science in Mathematics program at the University of Montpellier.

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

Hourly volumes:

            CM: 9:00 p.m.

            TD: 9:00 p.m.

            TP: 0

            Land: 0