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Fields and Galois Theory by J.S. Milne

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Fields and Galois Theory written by J.S. Milne . This is an other great mathematics book cover the following topics.

  • Basic Definitions and Results
    Rings, Fields, The characteristic of a field, Review of polynomial rings, Factoring polynomials, Extension fields, The subring generated by a subset, The subfield generated by a subset, Construction of some extension fields, Stem fields, Algebraic and transcendental elements, Transcendental numbers, Constructions with straight-edge and compass, Algebraically closed fields, Exercises

  • Splitting Fields; Multiple Roots
    Maps from simple extensions, Splitting fields, Multiple roots, Exercises

  • The Fundamental Theorem of Galois Theory
    Groups of automorphisms of fields, Separable, normal, and Galois extensions, The fundamental theorem of Galois theory, Examples, Constructible numbers revisited, The Galois group of a polynomial, Solvability of equations, Exercises

  • Computing Galois Groups
    When is Gf  An?, When is Gf transitive?, Polynomials of degree at most three, Quartic polynomials, Examples of polynomials with Sp as Galois group over Q, Finite fields, Computing Galois groups over Q, Exercises

  • Applications of Galois Theory
    Primitive element theorem, Fundamental Theorem of Algebra, Cyclotomic extensions, Dedekind’s theorem on the independence of characters, The normal basis theorem, Hilbert’s Theorem 90, Cyclic extensions, Kummer theory, Proof of Galois’s solvability theorem, Symmetric polynomials, The general polynomial of degree n, Norms and traces, Exercises

  • Algebraic Closures
    Zorn’s lemma, First proof of the existence of algebraic closures, Second proof of the existence of algebraic closures, Third proof of the existence of algebraic closures, (Non)uniqueness of algebraic closures, Separable closures

  • Infinite Galois Extensions
    Topological groups, The Krull topology on the Galois group, The fundamental theorem of infinite Galois theory, Galois groups as inverse limits, Nonopen subgroups of finite index

  • The Galois theory of etale algebras ´
    Review of commutative algebra, Etale algebras over a field, Classification of etale algebras over a field, Comparison with the theory of covering spaces

  • Transcendental Extensions
    Algebraic independence, Transcendence bases, Luroth’s theorem, Separating transcendence bases, Transcendental Galois theory
  • Exercises
    A Review Exercises, B Two-hour Examination, C Solutions to the Exercises
  • Index

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