Galois Theory-a first course

Apr 12, 2018 - morphism σ may send some of the coefficients of h – including the ..... Show that neither of these fields contain a proper subfield (hint: for Fp, ...
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Galois Theory – a first course

arXiv:1804.04657v1 [math.GR] 12 Apr 2018

Brent Everitt⋆

Contents 0. What is Galois Theory? . . . . . . . . . . . . . . . 1. Rings I: Polynomials . . . . . . . . . . . . . . . . 2. Roots and Irreducibility . . . . . . . . . . . . . . . 3. Fields I: Basics, Extensions and Concrete Examples 4. Rings II: Quotients . . . . . . . . . . . . . . . . . 5. Fields II: Constructions and More Examples . . . . 6. Ruler and Compass Constructions I . . . . . . . . . 7. Vector Spaces I: Dimensions . . . . . . . . . . . . 8. Fields III: Splitting Fields and Finite Fields . . . . 9. Ruler and Compass Constructions II . . . . . . . . 10. Groups I: Soluble Groups and Simple Groups . . . 11. Groups II: Symmetries of Fields . . . . . . . . . . 12. Vector Spaces II: Solving Equations . . . . . . . . 13. The Fundamental Theorem of Galois Theory . . . . 14. Applications of the Galois Correspondence . . . . 15. (Not) Solving Equations . . . . . . . . . . . . . .

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Introductory Note These notes are a self-contained introduction to Galois theory, designed for the student who has done a first course in abstract algebra. To not clutter up the theorems too much, I have made some restrictions in generality. For example, all rings are with 1; all ideals are principal; all fields are perfect – in fact, extensions Brent Everitt: Department of Mathematics, University of York, York YO10 5DD, United Kingdom. e-mail: [email protected]

version April 16, 2018.

2

Brent Everitt

of Q or of finite fields; consequently all field extensions are separable; and so on. This won’t be to everyone’s taste. The following prerequisites are assumed, although there are reminders: the basics of linear algebra, particularly the span and independence of a set of vectors; the idea of a basis and hence the dimension of a vector space. In group theory the fundamentals upto Lagrange’s theorem and the first isomorphism theorem. In ring and field theory the definitions and some examples, but probably not much else. There are many books on linear algebra and group theory for beginners. My personal favourite is: [Arm88] M. A. Armstrong, Groups and symmetry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1988. MR965514

Most of the results and proofs are standard and can be found in any book on Galois theory, but I am particularly indebted to the book of Joseph Rotman: [Rot90] Joseph Rotman, Galois theory, Universitext, Springer-Verlag, New York, 1990. MR1064318

In particular the proofs I give of Theorems C and E, the Fundamental Theorem of Algebra and the Theorem of Abel