Notes on Galois Theory - IIT Bombay

1 Notes on Galois TheorySudhir R. GhorpadeDepartment of Mathematics, Indian Institute of Technology, Bombay 400 076E-mail : 1994 Contents1 Preamble22 Field Extensions33 Splitting Fields and Normal Extensions64 Separable Extensions95 Galois Theory116 Norms and Traces1611 PreambleThese Notes attempt to give an introduction to some basic aspects of Field Theory and GaloisTheory. Originally, the succeeding sections of these notesconstituted a part of the notesprepared to supplement the lectures of the author on Galois Theory and Ramification Theoryat the All India Summer School in Number Theory held at Pune inJune 1991.

2 Subsequently,the first 6 sections of the Pune Notes were separated and slightly revised to form these Noteson Galois Theory , which were used for pre-conference distribution to the participants of theNBHM sponsored Instructional School on Algebraic Number Theory (University of Bombay ,December 1994) at the request of the organisers. A few minor revisions have taken place inthe subsequent main aim of these Notes has always been to provide a geodesic, yet complete, presen-tation starting from the definition of field extensions and concluding with the FundamentalTheorem of Galois Theory . Some additional material on separable extensions and a section onNorms and Traces is also included, and some historical comments appear as footnotes.

3 Theprerequisite for these Notes is basic knowledge of AbstractAlgebra and Linear Algebra notbeyond the contents of usual undergraduate courses in thesesubjects. No formal backgroundin Galois Theory is assumed. While a complete proof of the Fundamental Theorem of GaloisTheory is given here, we do not discuss further results such as Galois theorem on solvabilityof equations by radicals. An annotated list of references for Galois Theory appears at the endof Section 5. By way of references for the last section, viz.,Norms and Traces, we recommendVan der Waerden s Algebra (F. Ungar Pub. Co., 1949) and Zariski Samuel s CommutativeAlgebra, Vol.

4 1 (Springer-Verlag, 1975).It appears that over the years, these Notes are often used by students primarily interestedin Number Theory . Thus it may be pertinent to remark at the outset that the topics discussedin these Notes are very useful in the study of Algebraic Number Theory1. In order to derivemaximum benefit from these Notes , the students are advised toattempt all the Exercisesand fill the missing steps, if any, in the proofs given. The author would appreciate receivingcomments, suggestions and criticism regarding these fact, questions concerning integers alone, can sometimes be answered only with the help of field ex-tensions and certain algebraic objects associated to instance, Kummer showed that the equationXp+Yp=Zphas no integer solution for a class of odd primesp, called regular primes, which include all oddprimes less than 100 except 37, 59 and 67.

5 Even a convenient definition of regular primes, not to mention theproof of Kummer s Theorem, involves many of the algebraic notions discussed in these lectures. Indeed, anodd prime isregularif it doesn t divide the class number of the cyclotomic field extension|Q( p) of|Q. Fordetails, see H. Edwards Springer monograph Fermat s LastTheorem (1977).22 Field ExtensionsLetKbe a field2. By a(field) extensionofKwe mean a field containingKas a a fieldLbe an extension ofK(we usually express this by saying thatL/K[read:LoverK] is an extension). ThenLcan be considered as a vector space overK. ThedegreeofLoverK, denoted by [L:K], is defined as[L:K] = dimKL= the vector space dimension [L:K]< , we say thatLis afinite extensionofKor thatLisfiniteoverK.

6 A subfieldKof IC such that [K:|Q]< is called analgebraic number fieldor simply anumber 1:Finite over finite is finite. More precisely, ifL/EandE/Kare field extensions,thenL is f inite over K L is f inite over E and E is f inite over Kand, in this case,[L:K] = [L:E][E:K].Proof:The implication is obvious. The rest follows easily from the observation thatif{ui}is anE basis ofLand{vj}is aK basis ofE, then{uivj}is aK basis a field extension. An element Lis said to bealgebraicoverKif it satisfiesa nonzero polynomial with coefficients inK, , 06=f(X) K[X] such thatf( ) = Lwhich is algebraic overK, we can find a monic polynomial inK[X] of leastpossible degree, satisfied by.

7 This is unique and is called theminimal polynomialof overK. It is easily seen to be irreducible and we will denote it by Irr( , K). Note that iff(X)is any monic irreducible polynomial satisfied by , then we must havef(X) =Irr( , K) andthat it generates the ideal{g(X) K[X] :g( ) = 0}inK[X].3 The extensionLofKis saidto bealgebraicif every element ofLis algebraic 2:Finite algebraic. That is, ifL/Kis a finite extension, then it is :For any L, there must exist a positive integernsuch that{1, , 2, .., n}islinearly dependent overK, thus showing that is algebraic 1:Show, by an example, that the converse of the above lemma is not true, now study extensions for which the are usually denoted byKorksince the German word for field is K orper.

8 Much of ModernField Theory was created by the German mathematician E. Steinitz; see his paper Algebraische Theorie derK orper , Crelle Journal (1910), pp. 167 308, for an original may be instructive to verify the observations made in the last few statements. General Hint: Use theDivision Algorithm inK[X].3 Definition:Given elements 1, .. , nin an extensionLof a fieldK, we defineK[ 1, .. , n] = the smallest subring ofLcontainingKand 1, .. , nK( 1, .. , n) = the smallest subfield ofLcontainingKand 1, .. , thatK[ 1, .. , n] precisely consists of elements of the formf( 1, .. , n) wheref(X1.)

9 , Xn) varies overK[X1, .. , Xn] (= the ring of polynomials in thenvariablesX1, .. , Xnwith coefficients inK) whereasK( 1, .. , n) precisely consists of elements of the formf( 1,.., n)g( 1,.., n)wheref(X1, .. , Xn), g(X1, .. , Xn) vary overK[X1, .. , Xn] withg( 1, .. , n)6= note thatK( 1, .. , n) is the quotient field ofK[ 1, .. , n] :An extensionLofKis said to befinitely generatedoverKif there exist 1, .. , ninLsuch thatL=K( 1, .. , n). We say thatLis asimpleextension ofKifL=K( ) for some simple extensions, the converse to Lemma 2 is true. In fact, we can say much 3:Let be an element in an overfieldLof a fieldK.

10 Then:K( )/K is algebraic is algebraic over K K[ ] =K( ) [K( ) :K]< .Moreover, if is algebraic overKandf(X) =Irr( , K), then there exists an isomorphism ofK( )ontoK[X]/(f(X))which maps toX, the residue class ofX, and the elements ofKto their residue :Without loss of generality, we can and will assume that 6= 0. The first assertiontrivially implies the second. Now, the map :K[X] Ldefined byf(X)7 f( ) is clearlya ring homomorphism whose image isK[ ]. If is algebraic overK, then the kernel of isa nonzero prime ideal inK[X] and is hence a maximal ideal (prove!). SoK[ ] K[X]/ker is a field containingKand.