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Fields and Galois Theory Fields and Galois Theory J.S. Milne Version 4.30, April 15, 2012 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. BibTeX information @misc{milneFT, author={Milne, James S.}, title={Fields and Galois Theory (v4.30)}, year={2012}, note={Available at www.jmilne.org/math/}, pages={124} } v2.01 (August 21, 1996). First version on the web. v2.02 (May 27, 1998). Fixed about 40 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. v3.01 (August 31, 2003). Fixed many minor errors; no change to num- bering; 99 pages. v4.00 (February 19, 2005). Minor corrections and improvements; added proofs to the section on infinite Galois theory; added material to the section on 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. v4.21 (September 28, 2008). Minor corrections; fixed problem with hyperlinks; 111 pages. v4.22 (March 30, 2011). Minor changes; changed TEXstyle; 126 pages. v4.30 (April 15, 2012). Minor fixes; added sections on etale´ algebras; 124 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. Copyright c 1996, 1998, 2002, 2003, 2005, 2008, 2011, 2012 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents3 Notations............................ 7 References........................... 7 1 Basic Definitions and Results 11 Rings............................. 11 Fields............................. 13 The characteristic of a field.................. 14 Review of polynomial rings ................. 15 Factoring polynomials.................... 17 Extension fields........................ 23 Construction of some extension fields............ 24 Stem fields .......................... 27 The subring generated by a subset.............. 27 The subfield generated by a subset.............. 29 Algebraic and transcendental elements............ 30 Transcendental numbers................... 32 Constructions with straight-edge and compass. 36 Algebraically closed fields.................. 41 Exercises........................... 43 2 Splitting Fields; Multiple Roots 45 Maps from simple extensions................. 45 Splitting fields ........................ 48 Multiple roots......................... 52 3 Exercises........................... 57 3 The Fundamental Theorem of Galois Theory 59 Groups of automorphisms of fields.............. 59 Separable, normal, and Galois extensions.......... 63 The fundamental theorem of Galois theory ......... 67 Examples........................... 73 Constructible numbers revisited............... 77 The Galois group of a polynomial.............. 79 Solvability of equations ................... 79 Exercises........................... 80 4 Computing Galois Groups 83 When is Gf An?..................... 83 When is Gf transitive? ................... 86 Polynomials of degree at most three............. 87 Quartic polynomials ..................... 88 Examples of polynomials with Sp as Galois group over Q . 92 Finite fields.......................... 95 Computing Galois groups over Q . 98 Exercises...........................102 5 Applications of Galois Theory 105 Primitive element theorem...................105 Fundamental Theorem of Algebra . 109 Cyclotomic extensions....................111 Dedekind’s theorem on the independence of characters . 116 The normal basis theorem ..................118 Hilbert’s Theorem 90.....................121 Cyclic extensions.......................125 Kummer theory........................128 4 Proof of Galois’s solvability theorem . 130 The general polynomial of degree n . 133 Norms and traces.......................139 Etale´ algebras.........................146 Exercises...........................150 6 Algebraic Closures 151 Zorn’s lemma.........................152 First proof of the existence of algebraic closures . 153 Second proof of the existence of algebraic closures . 154 Third proof of the existence of algebraic closures . 155 (Non)uniqueness of algebraic closures . 157 Separable closures ......................159 7 Infinite Galois Extensions 161 Topological groups......................161 The Krull topology on the Galois group . 164 The fundamental theorem of infinite Galois theory . 168 Galois groups as inverse limits . 173 Nonopen subgroups of finite index . 177 Etale´ algebras.........................179 8 Transcendental Extensions 181 Algebraic independence ...................181 Transcendence bases.....................184 Luroth’s¨ theorem.......................190 Separating transcendence bases . 190 Transcendental Galois theory . 192 A Review Exercises 195 5 B Two-hour Examination 207 C Solutions to the Exercises 209 Index 227 6 Notations. We use the standard (Bourbaki) notations: 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 con- taining . The cardinality of a set S is denoted by S (so S is the number of elements in S when S is finite). Let I jandj A bej j sets. A family of elements of A indexed by I , denoted .ai /i I , is a func- 2 tion i ai I A. Throughout the notes, p is a prime number: p 2;3;5;7;11;:::7! W ! . D X YX is a subset of Y (not necessarily proper). def X YX is defined to be Y , or equals Y by definition. X D YX is isomorphic to Y . X YX and Y are canonically isomorphic. ' PREREQUISITES Group theory (for example, GT), basic linear algebra, and some elemen- tary theory of rings. References. Dummit, D., and Foote, R.M., 1991, Abstract Algebra, Prentice Hall. 7 Jacobson, N., 1964, Lectures in Abstract Algebra, Volume III — Theory of Fields and Galois Theory, van Nostrand. Also, the following of my notes (available at www.jmilne.org/math/). GT Group Theory, v3.12, 2012. ANT Algebraic Number Theory, v3.03, 2011. CA A Primer of Commutative Algebra, v2.23, 2012. A reference monnnnn is to http://mathoverflow.net/questions/nnnnn/ PARI is an open source computer algebra system freely available from http://pari.math.u-bordeaux.fr/. ACKNOWLEDGEMENTS I thank the following for providing corrections and comments for earlier versions of the notes: Mike Albert, Maren Baumann, Leendert Blei- jenga, Tommaso Centeleghe, Sergio Chouhy, Demetres Christofides, Antoine Chambert-Loir, Dustin Clausen, Keith Conrad, Hardy Falk, Jens Hansen, Albrecht Hess, Philip Horowitz, Trevor Jarvis, Henry Kim, Martin Klazar, Jasper Loy Jiabao, Dmitry Lyubshin, John McKay, Georges E. Melki, Courtney Mewton, Shuichi Otsuka, Dmitri Panov, Alain Pichereau, David G. Radcliffe, Roberto La Scala, Chad Schoen, Prem L Sharma, Dror Speiser, Bhupendra Nath Tiwari, Mathieu Vien- ney, Martin Ward (and class), Xiande Yang, and others. 8 Chapter 1 Basic Definitions and Results Rings A ring is a set R with two composition laws 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 a R D D 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 1We follow Bourbaki in requiring that rings have a 1, which entails that we require homomorphisms to preserve it. 11 12 1. BASIC DEFINITIONS AND RESULTS We usually omit “ ” and write 1 for 1R when this causes no confusion. If 1R 0, then R 0 . D D f g A subring S of a ring R is a subset 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 with the properties W ! 0 ˛.a b/ ˛.a/ ˛.b/ all a;b R; C D C 2 ˛.ab/ ˛.a/˛.b/ all a;b R; D 2 ˛.1R/ 1R : D 0 A ring R is said to be commutative if multiplication is commutative: ab ba for all a;b R: D 2 A commutative ring is said to be an integral domain if 1R 0 and the cancellation law holds for multiplication: ¤ ab ac, a 0, implies b c: D ¤ D An ideal I in a commutative ring R is a subgroup of .R; / that is closed under multiplication by elements of R: C r R, a I , implies ra I: 2 2 2 The ideal generated by elements a1;:::;an is denoted .a1;:::;an/. For example, .a/ is the principal ideal aR. We assume that the reader has some familiarity with the elementary theory of rings. For example, in Z (more generally, any Euclidean domain) an ideal I is generated by any “smallest” nonzero element of I . Fields 13 Fields DEFINITION 1.1 A field is a set F with two composition laws and such that C (a) .F; / is a commutative group; C (b) .F ; /, where F F 0 , is a commutative group; D X f g (c) the distributive law holds. Thus, a field is a nonzero commutative ring such that every nonzero element has an inverse. In particular, it is an integral domain. A field contains at least two distinct elements, 0 and 1. The smallest, and one of the most important, fields is F2 Z=2Z 0;1 . A subfield S of a field F is a subringD thatD isf closedg under passage to the inverse. It inherits the structure of a field from that on F . LEMMA 1.2 A nonzero commutative ring R is a field if and only if it has no ideals other than .0/ and R. PROOF. Suppose R is a field, and let I be a nonzero ideal in 1 R. If a is a nonzero element of I , then 1 a a I , and so I R. Conversely, suppose R is a commutativeD ring2 with no nontrivialD ideals. If a 0, then .a/ R, and so there exists a b ¤ D in R such that ab 1. 2 D EXAMPLE 1.3 The following are fields: Q, R, C, Fp Z=pZ (p prime): D A homomorphism of fields ˛ F F 0 is simply a homomorphism of rings.
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