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Fields and Galois Theory Fields and Galois Theory J.S. Milne Version 4.21 September 28, 2008 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. They are an expansion of those handed out during a course taught to first-year graduate students. BibTeX information @misc{milneFT, author={Milne, James S.}, title={Fields and Galois Theory (v4.21)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={107+iv} } 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 numbering; 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. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of Sabre Peak, Moraine Creek, New Zealand. Copyright c 1996, 1998, 2002, 2003, 2005, 2008, J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Notations. 3; References. 3 1 Basic definitions and results 1 Rings 1; Fields 1; The characteristic of a field 2; Review of polynomial rings 3; Factoring polynomials 4; Extension fields 7; Construction of some extension fields 8; Stem fields 9; The subring generated by a subset 9; The subfield generated by a subset 10; Algebraic and transcendental elements 11; Transcendental numbers 12; Constructions with straight-edge and compass. 14; Algebraically closed fields 17; Exercises 18 2 Splitting fields; multiple roots 19 Maps from simple extensions. 19; Splitting fields 20; Multiple roots 22; Exercises 24 3 The fundamental theorem of Galois theory 25 Groups of automorphisms of fields 25; Separable, normal, and Galois extensions 27; The fundamental theorem of Galois theory 29; Examples 31; Constructible numbers revisited 33; The Galois group of a polynomial 34; Solvability of equations 34; Exercises 34 4 Computing Galois groups. 36 When is G An? 36; When is G transitive? 37; Polynomials of degree at most three f f 37; Quartic polynomials 38; Examples of polynomials with Sp as Galois group over Q 40; Finite fields 41; Computing Galois groups over Q 43; Exercises 45 5 Applications of Galois theory 47 Primitive element theorem. 47; Fundamental Theorem of Algebra 49; Cyclotomic exten- sions 50; Dedekind’s theorem on the independence of characters 52; The normal basis theorem 53; Hilbert’s Theorem 90. 55; Cyclic extensions. 57; Kummer theory 58; Proof of Galois’s solvability theorem 60; The general polynomial of degree n 61; Norms and traces 64; Exercises 68 6 Algebraic closures 69 Zorn’s lemma 69; First proof of the existence of algebraic closures 70; Second proof of the existence of algebraic closures 70; Third proof of the existence of algebraic closures 70; (Non)uniqueness of algebraic closures 72; Separable closures 72 7 Infinite Galois extensions 74 Topological groups 74; The Krull topology on the Galois group 75; The fundamental the- orem of infinite Galois theory 77; Galois groups as inverse limits 80; Nonopen subgroups of finite index 81 8 Transcendental extensions 83 Algebraic independence 83; Transcendence bases 84;Luroth’s¨ theorem 87; Separating transcendence bases 87; Transcendental Galois theory 88 A Review exercises 90 B Two-hour Examination 95 C Solutions to the Exercises 96 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 containing . The cardi- nality of a set S is denoted by S (so S is the number of elements in S when S is finite). j j j j Let I and A be sets. A family of elements of A indexed by I , denoted .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! X W YX! is a subset of Y (not necessarily proper). D X def 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. Dummit, D., and Foote, R.M., 1991, Abstract Algebra, Prentice Hall. 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, v3.00, 2007. ANT Algebraic Number Theory, v3.00, 2008. Acknowledgements I thank the following for providing corrections and comments for earlier versions of the notes: Mike Albert, Maren Baumann, Leendert Bleijenga, Tommaso Centeleghe, Deme- tres 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, Courtney Mewton, Shuichi Otsuka, David G. Radcliffe, Dror Speiser, Mathieu Vienney, Martin Ward (and class), Xiande YANG, and others. PARI is an open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. 1 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 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 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/; ˛.ab/ ˛.a/˛.b/; ˛.1R/ 1R ; all a; b R: C D C D D 0 2 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 multipli- C cation by elements of R: 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 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 r 0 , is a commutative group; D f g (c) the distributive law holds. 1We follow Bourbaki in requiring that rings have a 1, which entails that we require homomorphisms to preserve it. 2 1 BASIC DEFINITIONS AND RESULTS 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 . D D f g A subfield S of a field F is a subring that is closed 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 R. If a is a nonzero element of I , then 1 a 1a I , and so I R. Conversely, suppose R is a commutative ring D 2 D with no nontrivial ideals. If a 0, then .a/ R, and so there exists a b in R such that ¤ D 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 is simply a homomorphism of rings. Such a W ! 0 homomorphism is always injective, because its kernel is a proper ideal (it doesn’t contain 1), which must therefore be zero. The characteristic of a field One checks easily that the map Z F; n 1F 1F 1F .n copies/; ! 7! C C C is a homomorphism of rings, and so its kernel is an ideal in Z.
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