Fields and Galois Theory J.S. Milne Œ Q QŒ;˛ 3 2 h i h i N H QŒ QŒp 7 QŒ QŒ˛ C x = 3 = 2 G=N h i h i h i h i Q Q 5 7 Splitting field of X 2 over . Splitting field of X 1 over Q. Q Version 5.00 June 2021 These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental extensions. The first six chapters form a standard course, and the final three chapters are more advanced. BibTeX information @misc{milneFT, author={Milne, James S.}, title={Fields and Galois Theory (v5.00)}, year={2021}, note={Available at www.jmilne.org/math/}, pages={142} } Please send comments and corrections to me at jmilne at umich.edu. v2.01 (August 21, 1996). First version on the web. v2.02 (May 27, 1998). Fixed many minor errors; 57 pages. v3.00 (April 3, 2002). Revised notes; minor additions to text; added 82 exercises with solutions, an examination, and an index; 100 pages. v4.00 (February 19, 2005). Revised notes; added proofs for Infinite Galois Extensions; expanded Transcendental Extensions; 107 pages. v4.10 (January 22, 2008). Minor corrections and improvements; added proofs for Kummer theory; 111 pages. v4.20 (February 11, 2008). Replaced Maple with PARI; 111 pages. v4.30 (April 15, 2012). Minor fixes; added sections on etale´ algebras; 124 pages. v4.50 (March 18, 2014). Added chapter on the Galois theory of etale´ algebras (Chapter 8); other improvements; numbering has changed; 138 pages. v4.61 (April 2020). Minor fixes and additions; numbering little changed; 138 pages. v5.00 (June 2021). First version available (with its source code) under a Creative Commons licence. Version 5.0 is published under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). Licence information: https://creativecommons.org/licenses/by-nc-sa/4.0 Copyright ©1996–2021 J.S. Milne. Contents Contents3 Notation. ...................................... 6 References. ..................................... 6 1 Basic Definitions and Results7 Rings ........................................ 7 Fields ........................................ 8 The characteristic of a field............................. 8 Review of polynomial rings............................. 10 Factoring polynomials ............................... 11 Extensions...................................... 14 The subring generated by a subset ......................... 15 The subfield generated by a subset......................... 15 Construction of some extensions.......................... 16 Stem fields...................................... 17 Algebraic and transcendental elements....................... 18 Transcendental numbers .............................. 19 Constructions with straight-edge and compass.................... 21 Algebraically closed fields ............................. 24 Exercises ...................................... 25 2 Splitting Fields; Multiple Roots 27 Homomorphisms from simple extensions...................... 27 Splitting fields.................................... 28 Multiple roots.................................... 30 Exercises ...................................... 33 3 The Fundamental Theorem of Galois Theory 35 Groups of automorphisms of fields......................... 35 Separable, normal, and Galois extensions ..................... 37 The fundamental theorem of Galois theory..................... 39 Examples ...................................... 42 Constructible numbers revisited .......................... 43 The Galois group of a polynomial ......................... 44 Solvability of equations............................... 45 Exercises ...................................... 46 4 Computing Galois Groups 47 When is G An?................................. 47 f 3 When does Gf act transitively on the roots?.................... 48 Polynomials of degree at most three ........................ 49 Quartic polynomials................................. 49 Examples of polynomials with Sp as Galois group over Q . 51 Finite fields ..................................... 52 Computing Galois groups over Q .......................... 54 Exercises ...................................... 58 5 Applications of Galois Theory 59 Primitive element theorem.............................. 59 Fundamental Theorem of Algebra ......................... 61 Cyclotomic extensions ............................... 62 Dedekind’s theorem on the independence of characters .............. 65 The normal basis theorem.............................. 66 Hilbert’s Theorem 90................................ 69 Cyclic extensions.................................. 71 Kummer theory................................... 72 Proof of Galois’s solvability theorem........................ 74 Symmetric polynomials............................... 75 The general polynomial of degree n ........................ 78 Norms and traces .................................. 79 Exercises ...................................... 83 6 Algebraic Closures 85 Zorn’s lemma.................................... 85 First proof of the existence of algebraic closures.................. 86 Second proof of the existence of algebraic closures ................ 86 Third proof of the existence of algebraic closures ................. 87 (Non)uniqueness of algebraic closures....................... 88 Separable closures.................................. 88 7 Infinite Galois Extensions 91 Topological groups ................................. 91 The Krull topology on the Galois group...................... 92 The fundamental theorem of infinite Galois theory................. 95 Galois groups as inverse limits........................... 98 Nonopen subgroups of finite index......................... 99 Exercises ......................................100 8 The Galois theory of etale´ algebras 103 Review of commutative algebra...........................103 Etale´ algebras over a field..............................104 Classification of etale´ algebras over a field.....................106 Comparison with the theory of covering spaces...................110 9 Transcendental Extensions 111 Algebraic independence...............................111 Transcendence bases ................................112 Luroth’s¨ theorem ..................................115 Separating transcendence bases...........................118 4 Transcendental Galois theory............................119 Exercises ......................................119 A Review Exercises 121 B Two-hour Examination 127 C Solutions to the Exercises 129 Index 141 5 Notation. We use the standard (Bourbaki) notation: N 0;1;2;::: ; D f g Z ring of integers, D R field of real numbers, D C field of complex numbers, D Fp Z=pZ field with p elements, p a prime number. D D Given an equivalence relation, Œ denotes the equivalence class containing . The cardinality of a set S is denoted by S (so S is the number of elements in Swhen S is finite). Let I j j j j and A be sets. A family of elements of A indexed by I , denoted by .ai /i I , is a function 2 i ai I A. Throughout the notes, p is a prime number: p 2;3;5;7;11;:::. 7! XW !YX is a subset of Y (not necessarily proper). D def X YX is defined to be Y , or equals Y by definition. D X YX is isomorphic to Y . X YX and Y are canonically isomorphic (or there is a given or unique isomorphism). ' PREREQUISITES Group theory (for example, GT), basic linear algebra, and some elementary theory of rings. References. Jacobson, N., 1964, Lectures in Abstract Algebra, Volume III, van Nostrand. Also, the following of my notes (available at www.jmilne.org/math/). GT Group Theory, v4.00, 2021. ANT Algebraic Number Theory, v3.08, 2020. CA A Primer of Commutative Algebra, v4.03, 2020. monnnn Question nnnn on mathoverflow.net. PARI An open source computer algebra system that you can run in your browser. It is freely available here. ACKNOWLEDGEMENTS I thank the following for providing corrections and comments for earlier versions of the notes: Mike Albert, Terezakis Alexios, Carlos Alberto Ajila Loayza, Lior Bary-Soroker, Maren Baumann, Leendert Bleijenga, Jin Ce, Tommaso Centeleghe, Sergio Chouhy, Demetres Christofides, Antoine Chambert-Loir, Dustin Clausen, Keith Conrad, Daniel Duparc, Hardy Falk, Ralf Goertz, Le Minh Ha, Matin Hajian, Jens Hansen, Albrecht Hess, Tim Holzschuh, Philip Horowitz, Ivan Ip, Trevor Jarvis, Henry Kim, Martin Klazar, Jasper Loy Jiabao, Weiyi Liu, Dmitry Lyubshin, Geir Arne Magnussen, John McKay, Sarah Manski, Georges E. Melki, Courtney Mewton, C Nebula, Shuichi Otsuka, Dmitri Panov, Artem Pelenitsyn, Alain Pichereau, David G. Radcliffe, Roberto La Scala, Chad Schoen, Rene´ Schoof, Prem L Sharma, Dror Speiser, Sam Spiro, Bhupendra Nath Tiwari, Mathieu Vienney, Martin Ward (and class), Yervand Yeghiazarian, Xiande Yang, Wei Xu, and others. 6 CHAPTER 1 Basic Definitions and Results Rings A ring is a set R with two binary operations and such that C (a) .R; / is a commutative group; C 1 (b) is associative, and there exists an element 1R such that a 1R a 1R a for all D D a R 2 I (c) the distributive law holds: for all a;b;c R, 2 .a b/ c a c b c C D C a .b c/ a b a c. C D C We usually omit “ ” and write 1 for 1R when this causes no confusion. If 1R 0, then D R 0 . D f g A subring of a ring R is a subset S that contains 1R and is closed under addition, passage to the negative, and multiplication. It inherits the structure of a ring from that on R. A homomorphism of rings ˛ R R is a map such that W ! 0 ˛.a b/ ˛.a/ ˛.b/; ˛.ab/ ˛.a/˛.b/; ˛.1R/ 1R C D C D D 0 for all a;b R. A ring R is