Fields and Galois Theory Fall 2004 Professor Yu-Ru Liu CHRIS ALMOST Contents 1 Introduction 3 1.1 Motivation.......................................................3 1.2 Brief Review of Ring Theory............................................3 2 Field extensions 4 2.1 Degree of a Field Extention.............................................4 2.2 Algebraic and Transcendental Numbers.....................................5 2.3 Simple Extensions...................................................5 2.4 Algebraic Extensions.................................................6 3 Splitting Fields 7 3.1 Existence of splitting fields.............................................7 3.2 Uniqueness of the splitting field..........................................8 4 Separable Polynomials 9 4.1 Prime Fields......................................................9 4.2 Formal Derivative and Repeated Roots......................................9 4.3 Separable Polynomials................................................ 10 4.4 Perfect Fields...................................................... 11 5 Automorphism Groups 12 5.1 Automorphism Groups................................................ 12 5.2 Automorphism Groups of Polynomials...................................... 12 5.3 Fixed Fields....................................................... 13 6 Galois Extensions 13 6.1 Separable Extensions................................................. 13 6.2 Normal extensions.................................................. 14 6.3 Conjugates....................................................... 16 6.4 Galois Extensions................................................... 16 6.5 Artin’s Theorem.................................................... 17 1 2 CONTENTS 7 The Galois Correspondence 19 7.1 The Fundemental Theorem............................................. 19 7.2 Applications...................................................... 21 7.3 Brief Review of Group Theory........................................... 21 7.4 The Primitive Element Theorem.......................................... 23 8 Ruler and Compass Constructions 24 8.1 Constructible Points.................................................. 24 8.2 Constructible Numbers................................................ 25 8.3 Applications...................................................... 25 9 Cyclotomic Extensions 27 9.1 Cyclotomic Polynomials............................................... 27 9.2 Cyclotomic Fields................................................... 28 9.3 Abelian Extensions.................................................. 28 9.4 Constructible n-gons................................................. 30 10 Galois Groups of Polynomials 30 10.1 Discriminant...................................................... 30 10.2 Cubic Polynomials................................................... 31 10.3 Quartic Polynomials................................................. 31 11 Solvability by Radicals 33 11.1 Cardano’s Formula.................................................. 33 11.2 Solvable groups.................................................... 35 11.3 Cyclic Extensions................................................... 37 11.4 Radical Extensions.................................................. 38 11.5 Solving polynomials by Radicals.......................................... 39 11.6 Probabilistic Galois Theory............................................. 40 INTRODUCTION 3 1 Introduction Galois Theory is the interplay between fields and groups. 1.1 Motivation Consider the following historical problems. Construct an arbitrary regular n-gon using only a ruler and a compass. We know how to construct a triangle • and square, but what about 5-gon, etc.? Square the circle using only a ruler and compass (i.e. construct a square of area π). • Solve an arbirary polynomial using only algebraic means (i.e. plus, minus, times, divides, and nthroot). • The quadratic formula gives a solution for quadratic equations. Cubic and quartic equations can be solved 3 similarily. e.g. if x + px = q then È È r 3 2 r 3 2 3 q p q 3 q p q x = + + + + 2 27 4 2 − 27 4 For which quintic equations do we have radical solutions? If we know there is such a solution, what does • the solution look like? How can we solve these problems? The main steps in applying the theory that we develope in this course are as follows: 1. Associate the solution of interest, say α = pπ or α = the root of some quintic, with the field Q(α). 2. Associate α with the group of isomorphisms of α that fix , Aut α . If α is algebraic then Q( ) Q( ) Q Q(Q( )) Aut α is finite. If α is constructable then the order of Aut α is in certain forms. Q(Q( )) Q(Q( )) Hard Question: How many intermediate fields between Q and Q(α)? There is a 1-1 correspondence between the intermediate fields and the subgroups of Aut α (this is the Fundemental Theorem of Galois theory.) Q(Q( )) 1.2 Brief Review of Ring Theory For this course we will be dealing with commutative rings with identity. 1.1 Example. Let R be a ring. We denote by R[x] the polynomial ring over R in indeterminant x. The degree of a polynomial is the exponent on the leading term. By convention, deg 0 = . If a polynomial has leading coefficient 1 then it is called “monic”. −∞ A ring R is called a domain if it has no zero divisors. An element u R is called a unit if it is invertible. A field is a commutative ring in which each non-zero element is a unit and 0 2= 1. 6 1.2 Example. If F is a field, then F[x] is a domain (it is sufficient that F be a domain) and for f , g F[x], deg(f g) = deg(f ) + deg(g). This degree function actually makes F[x] into a Euclidean domain. 2 The rational (function) field over a field F is denoted F(x) and consists of all quotients of polynomials (with non-zero denominator) from F[x]. It is the smallest field that contains F[x]. An ideal I of a ring R is a (not necessarily unital) subring of R that is absorbing with respect to multiplication by elements of R. We can now construct R=I, the quotient ring modulo I. I is said to be maximal if I = R and for any ideal J we have I J R I = J J = R. I is said to be prime if I = R and ab I a I b6 I. Notice that every maximal ideal⊆ ⊆ is prime,) and_ in PIDs every prime ideal is maximal.6 Fields2 have) only2 trivial_ 2 ideals. 4 FIELDSAND GALOIS 1.3 Theorem. Let I be a proper ideal of R. Then 1. R=I is a field if and only if I is maximal 2. R=I is a domain if and only if I is prime 1.4 Theorem. (First Isomorphism Theorem) If ' : R S is a ring homomorphism and ker ' = I then there is an isomorphism ! α : R=I Im ' : r + I '(r) ! 7! 2 Field extensions 2.1 Definition. If E is a field containing another field F then E is said to be a field extension of F, denoted by E=F 2.1 Degree of a Field Extention If E=F is a field extension then we can view E as a vector space over F. Addition is given to agree with the field addition • Scalar multiplication is given to agree with the field multiplication • 2.2 Definition. The dimension of E viewed as a vector space over F is called the degree of E over F and is denoted [E : F]. If this quantity happens to be finite, then E=F is said to be a finite extension, otherwise it is an infinite extension. 2.3 Example. 1. C = R iR, so [C : R] = 2 ∼ ⊕ 2. [R : Q] = 1 1 2 3. Let F be a field. The rational field is an infinite extension. An infinite linearly independent set is ..., x − , 1, x, x ,... f g 2.4 Theorem. If E=K and K=F are finite field extensions, then E=F is finite and [E : F] = [E : K][K : F] PROOF: Let a1,..., am be a basis for E over K and b1,..., bn be a basis for K over F. It suffices to prove f g f g α := ai bj 1 i m, 1 j n is a basis for E over F. Every element of E is a linear combination of elements of α fsincej each≤ element≤ ≤ of E≤isg a linear combination of elements of a1,..., am , and each of the ai’s (being elements of K) can be written as a linear combination of elements fromf b1,..., gbn . α is linearly independent Pm Pn f g Pn over F, for otherwise if i 1 j 1 ci,j bj ai = 0, then a1,..., am a basis implies that j 1 ci,j bj = 0 for all i. = = f g = Since b1,..., bn is also a basis, we get that ci,j = 0 for all i and j. f g 2.5 Definition. Let E=F be a field extension. If K is a subfield of E that contains F then we say that K is an intermediate field of E=F. 2.6 Corollary. If E=F is a finite extension and K is an intermediate field then [E : K] and [K : F] are divisors of [E : F]. FIELDEXTENSIONS 5 2.2 Algebraic and Transcendental Numbers 2.7 Definition. Let E=F be a field extension and α E. We say that α is algebraic over F if there is f (x) F[x] such that f = 0 and f (α) = 0. Otherwise α is said to2 be transcendental over F. 2 6 In particular, for α C and α algebraic (transcendental) over Q, we say that α is an algebraic (transcendental) 3 number. For example,2 all rational numbers are algebraic, as are p2, p2 + i, etc. The real numbers e (Hermite 1873) and π (Lindemann 1882) are transcendental numbers. 2.8 Theorem. (Liouville 1884) Let α R Q be a root of a polynomial f (x) Q[x] of degree n. Then there exists a constant c > 0 such that for any2 rationaln number p with q > 0 2 q p c α > n − q q p PROOF: Without loss of generality, we can assume α < 1 and that f x x and f is irreducible. Then q ( ) Z[ ] p j − pj p 2 p f α 0 and f 0. By the Mean Value theorem, f f α f M α , where M sup f x ( ) = ( q ) = ( q ) = ( ) ( q ) q = 0( ) 6 j j j − j ≤p j − j p j j for x α < 1. Since α is irrational, deg f 2 and M 0. Furthermore, f 1=qn, and thus α 1 1 , ( ) = ( q ) q M qn j − j ≥ 6 j j ≥ j − j ≥ so take c 1 . = M Remark. Liouville’s Theorem says that algebraic numbers are “harder” to approximate by rational numbers than transcendental numbers.