Milne J.S. Fields and Galois Theory

1996. - 57 pages.
The pioneering work of Abel and Galois in the early nineteenth century demonstrated that the long-standing quest for a solution of quintic equations by radicals was fruitless: no formula can be found. The techniques they used were, in the end, more important than the resolution of a somewhat esoteric problem, for they were the genesis of modern abstract algebra.* Extensions of Fields (Definitions, The characteristic of a field, The polynomial ring F[X], Factoring polynomials, Extension fields, Construction of some extensions, Generators of extension fields, Algebraic and transcendental elements, Transcendental numbers, Constructions with straight-edge and compass.
* Splitting Fields and Algebraic Closures (Maps from simple extensions, Splitting fields, Algebraic closures)
* The Fundamental Theorem of Galois Theory (Multiple roots, Groups of automorphisms of fields, Separable, normal, and Galois extensions, The fundamental theorem of Galois theory, Constructible numbers revisited, Galois group of a polynomial, Solvability of equations)
* Computing Galois Groups (When is Gf ⊂ An, When is Gf transitive, Polynomials of degree ≤3, Quartic polynomials, Examples of polynomials with Sp as Galois group over Q, Finite fields, Computing Galois groups over Q
* Applications of Galois Theory, Primitive element theorem, Fundamental Theorem of Algebra, Cyclotomic extensions, Independence of characters, Hilbert’s Theorem, Cyclic extensions, Proof of Galois’s solvability theorem, The general polynomial of degree n, Symmetric polynomials, The general polynomial, A brief history, Norms and traces, Infinite Galois extensions