To introduce the theory of polynomial equations of one variable: Galois Theory.

To introduce criteria when a polynomial equation can be solved in radicals, when a geometric construction can be performed by a ruler and a compass.

(LO1) Know why and how a polynomial equation of degree up to 4 can be solved in radicals.

(LO2) Understand why a solution in radicals is impossible in general for the degree greater than or equal to 5.

(LO3) Understand when a polynomial can be solved in radicals.

(LO4) Know when a geometric construction can be done by a ruler and compass.

(LO5) Know what is the Galois group of a polynomial which permits the above results.

Commutative rings, ideals, fields, the polynomial ring.

The ring of integers, its ideals, fields with p elements. Fields, the ring of polynomials, its ideals.Characteristic of a field. The splitting field of a polynomial. The algebraic closure. Finite fields. Finite algebraic extensions. Galois Theory.

Algebraic extensions. Degree of extension as dimension of a vector space. Separable, non-separable extensions. Normal extensions. Galois extensions. Galois group of a field extension and a polynomial. Main Galois Theorem. Normal subgroups and extensions. Classification of finite fields. Fundamental Theorem of Algebra. Examples on calculations of Galois groups. Solubility in radicals.

Radicals, cyclic groups and extensions. Roots of unity. Kummer theory. Solubility in radicals. Soluble groups. Simplicity of an alternating group of degree at least 5. The field of rational functions. The general polynomial equation. Symmetric functions. Solubility in radicals of an equ ation of degree less than 4. Non-solubility in radicals of a general equation of degree greater than 4. Geometric constructions by a ruler and compass.

Criterion of constructability by a ruler and compass. The duplication of a cube. The trisection of an angle. Regular polygons. Gauss Theorem.