MATH 456 Galois Theory

Cubic and quartic equations. Cardan's Formulas. Symmetric polynomials. Discriminant. Roots of polynomials. The Fundamental Theorem of Algebra. Extension fields. Minimal polynomials. Adjoining elements. Degree of a field extension. Finite extensions. The tower theorem. Algebraic extensions. Simple extensions. Splitting fields, their uniqueness up to isomorphism. Normal extensions. Separable extensions. Fields of characteristic 0 and fields of characteristic p. The Primitive Element Theorem. Galois group. Galois group of splitting fields. Permutation of the roots. Examples of Galois groups. Abelian equations. Galois extensions. The Fundamental Theorem of Galois Theory. Solvability by radicals. Solvable groups. Cyclotomic extensions. Regular polygons and roots of unity. Impossibility of some geometric constructions using just straightedge and compass. Finite fields.