• Field extensions, degree of an extension. Algebraic and transcendental element, minimum polynomial of an Algebraic element . Finite extensions. Algebraic extensions.
• Splitting fields of a polynomial. K-isomorphisms. Algebraic closure.
• Normal extension. Separable extension. The primitive element theorem. Trace, norm and discriminant.
• Finite fields, finite multiplicative sub-groups of a field. The structure of finite field. Roots of unity, cyclotomic polynomials. Wedderburn theorem.
Galois extensions, the galois group. The fundamental theorem of galois theory, applications. Ruler and compasses constructions. Solution of equations by radicals.