Algebra 2

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

To master the basic tools of commutative algebra, and to introduce a typical correspondence theorem, very fruitful 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 University of 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 which 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 the roots, Galois theorem of resolvability by radicals in zero characteristic.

Hourly volumes:

            CM : 21h

            TD : 21h

            TP : 0

            Land : 0