Fields and Galois Theory - jmilne.org

1 Fields and Galois MilneVersion , April 15, 2012 These notes give a concise exposition of the Theory of Fields , includingthe Galois Theory of finite and infinite extensions and the Theory oftranscendental information@misc{milneFT,author={Milne, James S.},title={ Fields and Galois Theory ( )},year={2012},note={Available at },pages={124}} (August 21, 1996). First version on the (May 27, 1998). Fixed about 40 minor errors; 57 (April 3, 2002). Revised notes; minor additions to text; added82 exercises with solutions, an examination, and an index; (August 31, 2003).

2 Fixed many minor errors; no change to num-bering; 99 (February 19, 2005). Minor corrections and improvements; addedproofs to the section on infinite Galois Theory ; added material tothe section on transcendental extensions; 107 (January 22, 2008). Minor corrections and improvements; addedproofs for Kummer Theory ; 111 (February 11, 2008). Replaced Maple with PARI; 111 (September 28, 2008). Minor corrections; fixed problem withhyperlinks; 111 (March 30, 2011). Minor changes; changed TEXstyle; 126 (April 15, 2012). Minor fixes; added sections on etale algebras;124 at send comments and corrections to me at the address on my 1996, 1998, 2002, 2003, 2005, 2008, 2011, 2012 paper copies for noncommercial personal use may be madewithout explicit permission from the copyright.

3 7 References..71 Basic Definitions and Results11 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 ..432 Splitting Fields ; Multiple Roots45 Maps from simple extensions.

4 45 Splitting Fields ..48 Multiple roots ..523 Exercises ..573 The Fundamental Theorem of Galois Theory59 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 ..804 Computing Galois Groups83 When isGf An? ..83 When isGftransitive? ..86 Polynomials of degree at most three ..87 Quartic polynomials.

5 88 Examples of polynomials withSpas Galois group Fields ..95 Computing Galois groups overQ..98 Exercises .. 1025 Applications of Galois Theory105 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 .. 1284 Proof of Galois s solvability theorem .. 130 The general polynomial of degreen.. 133 Norms and traces.

6 139 Etale algebras .. 146 Exercises .. 1506 Algebraic Closures151 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 .. 1597 Infinite Galois Extensions161 Topological groups .. 161 The Krull topology on the Galois group .. 164 The fundamental theorem of infinite Galois Theory .. 168 Galois groups as inverse limits.

7 173 Nonopen subgroups of finite index .. 177 Etale algebras .. 1798 Transcendental Extensions181 Algebraic independence .. 181 Transcendence bases .. 184L uroth s theorem .. 190 Separating transcendence bases .. 190 Transcendental Galois Theory .. 192 AReview Exercises1955 BTwo-hour Examination207 CSolutions to the use the standard (Bourbaki) notations:NDf0;1;2;:::g;ZDring of integers,RDfield of real numbers,CDfield of complex numbers,FpDZ=pZDfield withpelements,pa prime an equivalence relation, denotes the equivalence class con-taining.

8 The cardinality of a setSis denoted byjSj(sojSjis thenumber of elements inSwhenSis finite). LetIandAbe sets. Afamily of elements ofAindexed byI, , is a func-tioni7!aiWI!A. Throughout the notes,pis a prime number:pD2;3;5;7;11;:::.X Y Xis a subset ofY(not necessarily proper).XdefDY Xis defined to beY, or equalsYby Y Xis isomorphic 'Y XandYare canonically Theory (for example, GT), basic linear algebra, and some elemen-tary Theory of , D., and Foote, , 1991, Abstract Algebra, Prentice , N., 1964, Lectures in Abstract Algebra, Volume III Theoryof Fields and Galois Theory , van , the following of my notes (available at ).

9 GTGroup Theory , , Number Theory , , Primer of Commutative Algebra, , reference monnnnn is an open source computer algebra system freely available thank the following for providing corrections and comments for earlierversions 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, HenryKim, Martin Klazar, Jasper Loy Jiabao, Dmitry Lyubshin, John McKay,Georges E.

10 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 1 Basic Definitions and ResultsRingsAringis a setRwith two composition lawsCand such that(a).R;C/is a commutative group;(b) is associative, and there exists1an element1 Rsuch thata 1 RDaD1R afor alla2RI(c) the distributive law holds: for alla;b;c2R,.aCb/ cDa cCb ca .bCc/Da bCa follow Bourbaki in requiring that rings have a1, which entails that werequire homomorphisms to preserve BASICDEFINITIONS ANDRESULTSWe usually omit and write1for1 Rwhen this causes no , a ringRis a subset that contains1 Rand is closedunder addition, passage to the negative, and multiplication.