Algebra 2

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

Master the basic tools of commutative algebra, and introduce a typical correspondence theorem that is highly fertile 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" of the Licence de Mathématiques of the Université de Montpellier.

1. Revisions on rings, bodies; prime subbody, characteristic of a body, Frobenius morphism, factorization and Eisenstein criterion.
2. Body extensions: degree formula, algebraic extensions, algebraically closed bodies, algebraic closures, breaking bodies, decomposition bodies, extensions of body morphisms.
3. The Galois group; invariant subbodies, Artin's theorem.
4. Finite bodies: Galois group, sub-bodies, Galois correspondence.
5. Normal extensions, those that are finite are decomposition bodies.
6. Polynomials and separable extensions: definitions, composition of separable extensions, perfect bodies (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 of resolubility by radicals in characteristic zero.

Hourly volumes :

            CM: 21h

            TD: 21h

            TP: 0

            Land: 0