# Part II Galois Theory

Size: px
Start display at page:

Transcription

1 Part II Galois Theory Definitions Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Groups, Rings and Modules is essential Field extensions, tower law, algebraic extensions; irreducible polynomials and relation with simple algebraic extensions. Finite multiplicative subgroups of a field are cyclic. Existence and uniqueness of splitting fields. [6] Existence and uniqueness of algebraic closure. [1] Separability. Theorem of primitive element. Trace and norm. [3] Normal and Galois extensions, automorphic groups. Fundamental theorem of Galois theory. [3] Galois theory of finite fields. Reduction mod p. [2] Cyclotomic polynomials, Kummer theory, cyclic extensions. Symmetric functions. Galois theory of cubics and quartics. [4] Solubility by radicals. Insolubility of general quintic equations and other classical problems. [3] Artin s theorem on the subfield fixed by a finite group of automorphisms. Polynomial invariants of a finite group; examples. [2] 1

2 Contents II Galois Theory (Definitions) Contents 0 Introduction 3 1 Solving equations 4 2 Field extensions Field extensions Ruler and compass constructions K-homomorphisms and the Galois Group Splitting fields Algebraic closures Separable extensions Normal extensions The fundamental theorem of Galois theory Finite fields Solutions to polynomial equations Cyclotomic extensions Kummer extensions Radical extensions Solubility of groups, extensions and polynomials Insolubility of general equations of degree 5 or more Computational techniques Reduction mod p Trace, norm and discriminant

3 0 Introduction II Galois Theory (Definitions) 0 Introduction 3

4 1 Solving equations II Galois Theory (Definitions) 1 Solving equations 4

5 2 Field extensions II Galois Theory (Definitions) 2 Field extensions 2.1 Field extensions Definition (Field extension). A field extension is an inclusion of a field K L, where K inherits the algebraic operations from L. We also write this as L/K. Alternatively, we can define this by a injective homomorphism K L. We say L is an extension of K, and K is a subfield of L. Definition (Degree of field extension). The degree of L over K is [L : K] is the dimension of L as a vector space over K. The extension is finite if the degree is finite. Definition (Algebraic number). Let L/K be a field extension, α L. We define I α = {f K[t] : f(α) = 0} K[t] This is the set of polynomials for which α is a root. It is easy to show that I α is an ideal, since it is the kernel of the ring homomorphism K[t] L by g g(α). We say α is algebraic over K if I α 0. Otherwise, α is transcendental over K. We say L is algebraic over K if every element of L is algebraic. Definition (Minimal polynomial). Let L/K be a field extension, α L. The minimal polynomial of α over K is a monic polynomial P α such that I α = P α. Definition (Field generated by α). Let L/K be a field extension, α L. We define K(α) to be the smallest subfield of L containing K and α. We call K(α) the field generated by α over K. Definition (Field generated by elements). Let L/K be a field extension, α 1,, α n L. We define K(α 1,, α n ) to be the smallest subfield of L containing K and α 1,, α n. We call K(α 1,, α n ) the field generated by α 1,, α n over K. 2.2 Ruler and compass constructions Definition (Constructible points). Let S R 2 be a set of (usually finite) points in the plane. A ruler allows us to do the following: if P, Q S, then we can draw the line passing through P and Q. A compass allows us to do the following: if P, Q, Q S, then we can draw the circle with center at P and radius of length QQ. Any point R R 2 is 1-step constructible from S if R belongs to the intersection of two distinct lines or circles constructed from S using rulers and compasses. A point R R 2 is constructible from S if there is some R 1,, R n = R R 2 such that R i+1 is 1-step constructible from S {R 1,, R i } for each i. Definition (Field of S). Let S R 2 be finite. Define the field of S by Q(S) = Q({coordinates of points in S}) R, where we put in the x coordinate and y coordinate separately into the generating set. 5

6 2 Field extensions II Galois Theory (Definitions) 2.3 K-homomorphisms and the Galois Group Definition (K-homomorphism). Let L/K and L /K be field extensions. A K-homomorphism φ : L L is a ring homomorphism such that φ K = id, i.e. it fixes everything in K. We write Hom K (L, L ) for the set of all K-homomorphisms L L. A K-isomorphism is a K-homomorphism which is an isomorphism of rings. A K-automorphism is a K-isomorphism L L. We write Aut K (L) for the set of all K-automorphism L L. Definition (Galois extension). Let L/K be a finite field extension. This is a Galois extension if Aut K (L) = [L : K]. Definition (Galois group). The Galois group of a Galois extension L/K is defined as Gal(L/K) = Aut K (L). The group operation is defined by function composition. It is easy to see that this is indeed a group. 2.4 Splitting fields Notation. Let L/K be a field extension, f K[t]. We write Root f (L) for the roots of f in L. Definition (Splitting field). Let L/K be a field extensions, f K[t]. We say f splits over L if we can factor f as f = a(t α 1 ) (t α n ) for some a K and α j L. Alternatively, this says that L contains all roots of f. We say L is a splitting field of f if L = K(α 1,, α n ). This is the smallest field where f has all its roots. 2.5 Algebraic closures Definition (Algebraically closed field). A field L is algebraically closed if for all f L[t], we have f = a(t α 1 )(t α 2 ) (t α n ) for some a, α i L. In other words, L contains all roots of its polynomials. Let L/K be a field extension. We say L is an algebraic closure of K if L is algebraic over K L is algebraically closed. 2.6 Separable extensions Definition (Separable polynomial). Let K be a field, f K[t] non-zero, and L a splitting field of f. For an irreducible f, we say it is separable if f has no repeated roots, i.e. Root f (L) = deg f. For a general polynomial f, we say it is separable if all its irreducible factors in K[t] are separable. 6

7 2 Field extensions II Galois Theory (Definitions) Definition (Formal derivative). Let K be a field, f K[t]. (Formal) differentiation the K-linear map K[t] K[t] defined by t n nt n 1. The image of a polynomial f is the derivative of f, written f. Definition (Separable elements and extensions). Let K L be an algebraic field extension. We say α L is separable over K if P α is separable, where P α is the minimal polynomial of α over K. We say L is separable over K (or L L is separable) if all α L are separable. 2.7 Normal extensions Definition (Normal extension). Let K L be an algebraic extension. We say L/K is normal if for all α L, the minimal polynomial of α over K splits over L. 2.8 The fundamental theorem of Galois theory Definition (Fixed field). Let L/K be a field extension, H Aut K (L) a subgroup. We define the fixed field of H as L H = {α L : φ(α) = α for all φ H}. It is easy to see that L H is an intermediate field K L H L. 2.9 Finite fields Definition. Consider the extension F q n/f q, where q is a power of p. Frobenius Fr q : F q n F q n is defined by α α q. The 7

8 3 Solutions to polynomial equations II Galois Theory (Definitions) 3 Solutions to polynomial equations 3.1 Cyclotomic extensions Definition (Cyclotomic extension). For a field K, we define the nth cyclotomic extension to be the splitting field of t n 1. Definition (Primitive root of unity). The nth primitive root of unity is an element of order n in Root t n 1(L). 3.2 Kummer extensions Definition (Cyclic extension). We say a Galois extension L/K is cyclic is Gal(L/K) is a cyclic group. Definition (Kummer extension). Let K be a field, λ K non-zero, n N, char K = 0 or 0 < char K n. Suppose K contains an nth primitive root of unity, and L is a splitting field of t n λ. If deg[l : K] = n, we say L/K is a Kummer extension. 3.3 Radical extensions Definition (Radical extension). A field extension L/K is radical if there is some further extension E/L and with a sequence K = E 0 E 1 E r = E, such that each E i E i+1 is a cyclotomic or Kummer extension, i.e. E i+1 is a splitting field of t n λ i+1 over E i for some λ i+1 E i. Definition (Solubility by radicals). Let K be a field, and f K[t]. f. We say f is soluble by radicals if the splitting field of f is a radical extension of K. 3.4 Solubility of groups, extensions and polynomials Definition (Soluble group). A finite group G is soluble if there exists a sequence of subgroups G r = {1} G 1 G 0 = G, where G i+1 is normal in G i and G i /G i+1 is cyclic. Definition (Soluble extension). A finite field extension L/K is soluble if there is some extension L E such that K E is Galois and Gal(E/K) is soluble. 3.5 Insolubility of general equations of degree 5 or more Definition (Field of symmetric rational functions). Let K be a field, L = K(x 1,, x n ), the field of rational functions over K. Then there is an injective homomorphism S n Aut K (L) given by permutations of x i. We define the field of symmetric rational functions F = L Sn to be the fixed field of S n. 8

9 3 Solutions to polynomial equations II Galois Theory (Definitions) Definition (Elementary symmetric polynomials). The elementary symmetric polynomials are e 1, e 2,, e n defined by e i = x l1 x li. 1 l 1<l 2< <l i n Definition (General polynomial). Let K be a field, u 1,, u n variables. The general polynomial over K of degree n is f = t n + u 1 t n u n. Technically, this is a polynomial in the polynomial ring K(u 1,, u n )[t]. However, we say this is the general polynomial over K be cause we tend to think of these u i as representing actual elements of K. 9

10 4 Computational techniques II Galois Theory (Definitions) 4 Computational techniques 4.1 Reduction mod p 4.2 Trace, norm and discriminant Definition (Trace). Let K be a field. If A = [a ij ] is an n n matrix over K, we define the trace of A to be tr(a) = n a ii, i=1 i.e. we take the sum of the diagonal terms. Definition (Trace of linear map). Let V be a finite-dimensional vector space over K, and σ : V V a K-linear map. Then we can define tr(σ) = tr(any matrix representing σ). Definition (Trace of element). Let K L be a finite field extension, and α L. Consider the K-linear map σ : L L given by multiplication with α, i.e. β αβ. Then we define the trace of α to be tr L/K (α) = tr(σ). Definition (Norm of element). We define the norm of α to be N L/K (α) = det(σ), where σ is, again, the multiplication-by-α map. Definition (Discriminant). Let K be a field and f K[t], L the splitting field of f over K. So we have for some a, α 1,, α n L. We define f = a(t α 1 ) (t α n ) f = (α i α j ), D f = 2 f = ( 1) n(n 1)/2 i α j ). i<j i j(α We call D f the discriminant of f. 10

### Part II Galois Theory

Part II Galois Theory Theorems Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

### Part II Galois Theory

Part II Galois Theory Based on lectures by C. Birkar Notes taken by Dexter Chua Michaelmas 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.

### Galois theory (Part II)( ) Example Sheet 1

Galois theory (Part II)(2015 2016) Example Sheet 1 c.birkar@dpmms.cam.ac.uk (1) Find the minimal polynomial of 2 + 3 over Q. (2) Let K L be a finite field extension such that [L : K] is prime. Show that

### Math 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille

Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is

### GALOIS THEORY. Contents

GALOIS THEORY MARIUS VAN DER PUT & JAAP TOP Contents 1. Basic definitions 1 1.1. Exercises 2 2. Solving polynomial equations 2 2.1. Exercises 4 3. Galois extensions and examples 4 3.1. Exercises. 6 4.

### Galois Theory, summary

Galois Theory, summary Chapter 11 11.1. UFD, definition. Any two elements have gcd 11.2 PID. Every PID is a UFD. There are UFD s which are not PID s (example F [x, y]). 11.3 ED. Every ED is a PID (and

### Contradiction. Theorem 1.9. (Artin) Let G be a finite group of automorphisms of E and F = E G the fixed field of G. Then [E : F ] G.

1. Galois Theory 1.1. A homomorphism of fields F F is simply a homomorphism of rings. Such a homomorphism is always injective, because its kernel is a proper ideal (it doesnt contain 1), which must therefore

### FIELD THEORY. Contents

FIELD THEORY MATH 552 Contents 1. Algebraic Extensions 1 1.1. Finite and Algebraic Extensions 1 1.2. Algebraic Closure 5 1.3. Splitting Fields 7 1.4. Separable Extensions 8 1.5. Inseparable Extensions

### Galois Theory TCU Graduate Student Seminar George Gilbert October 2015

Galois Theory TCU Graduate Student Seminar George Gilbert October 201 The coefficients of a polynomial are symmetric functions of the roots {α i }: fx) = x n s 1 x n 1 + s 2 x n 2 + + 1) n s n, where s

### The Galois group of a polynomial f(x) K[x] is the Galois group of E over K where E is a splitting field for f(x) over K.

The third exam will be on Monday, April 9, 013. The syllabus for Exam III is sections 1 3 of Chapter 10. Some of the main examples and facts from this material are listed below. If F is an extension field

### Math 603, Spring 2003, HW 6, due 4/21/2003

Math 603, Spring 2003, HW 6, due 4/21/2003 Part A AI) If k is a field and f k[t ], suppose f has degree n and has n distinct roots α 1,..., α n in some extension of k. Write Ω = k(α 1,..., α n ) for the

### Fields. Victoria Noquez. March 19, 2009

Fields Victoria Noquez March 19, 2009 5.1 Basics Definition 1. A field K is a commutative non-zero ring (0 1) such that any x K, x 0, has a unique inverse x 1 such that xx 1 = x 1 x = 1. Definition 2.

### Field Theory Qual Review

Field Theory Qual Review Robert Won Prof. Rogalski 1 (Some) qual problems ˆ (Fall 2007, 5) Let F be a field of characteristic p and f F [x] a polynomial f(x) = i f ix i. Give necessary and sufficient conditions

### 1 The Galois Group of a Quadratic

Algebra Prelim Notes The Galois Group of a Polynomial Jason B. Hill University of Colorado at Boulder Throughout this set of notes, K will be the desired base field (usually Q or a finite field) and F

### Jean-Pierre Escofier. Galois Theory. Translated by Leila Schneps. With 48 Illustrations. Springer

Jean-Pierre Escofier Galois Theory Translated by Leila Schneps With 48 Illustrations Springer Preface v 1 Historical Aspects of the Resolution of Algebraic Equations 1 1.1 Approximating the Roots of an

### Math 201C Homework. Edward Burkard. g 1 (u) v + f 2(u) g 2 (u) v2 + + f n(u) a 2,k u k v a 1,k u k v + k=0. k=0 d

Math 201C Homework Edward Burkard 5.1. Field Extensions. 5. Fields and Galois Theory Exercise 5.1.7. If v is algebraic over K(u) for some u F and v is transcendental over K, then u is algebraic over K(v).

### but no smaller power is equal to one. polynomial is defined to be

13. Radical and Cyclic Extensions The main purpose of this section is to look at the Galois groups of x n a. The first case to consider is a = 1. Definition 13.1. Let K be a field. An element ω K is said

### School of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information

MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon

### Profinite Groups. Hendrik Lenstra. 1. Introduction

Profinite Groups Hendrik Lenstra 1. Introduction We begin informally with a motivation, relating profinite groups to the p-adic numbers. Let p be a prime number, and let Z p denote the ring of p-adic integers,

### Name: Solutions Final Exam

Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. 1. [10 Points] For

### Galois theory and the Abel-Ruffini theorem

Galois theory and the Abel-Ruffini theorem Bas Edixhoven November 4, 2013, Yogyakarta, UGM A lecture of two times 45 minutes. Audience: bachelor, master and PhD students, plus maybe some lecturers. This

### 1 Rings 1 RINGS 1. Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism

1 RINGS 1 1 Rings Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism (a) Given an element α R there is a unique homomorphism Φ : R[x] R which agrees with the map ϕ on constant polynomials

### Quasi-reducible Polynomials

Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let

### Fields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.

Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should

### RUDIMENTARY GALOIS THEORY

RUDIMENTARY GALOIS THEORY JACK LIANG Abstract. This paper introduces basic Galois Theory, primarily over fields with characteristic 0, beginning with polynomials and fields and ultimately relating the

### Course 311: Abstract Algebra Academic year

Course 311: Abstract Algebra Academic year 2007-08 D. R. Wilkins Copyright c David R. Wilkins 1997 2007 Contents 3 Introduction to Galois Theory 41 3.1 Field Extensions and the Tower Law..............

### Homework 4 Algebra. Joshua Ruiter. February 21, 2018

Homework 4 Algebra Joshua Ruiter February 21, 2018 Chapter V Proposition 0.1 (Exercise 20a). Let F L be a field extension and let x L be transcendental over F. Let K F be an intermediate field satisfying

### GALOIS THEORY AT WORK

GALOIS THEORY AT WORK KEITH CONRAD 1. Examples Example 1.1. The field extension Q(, 3)/Q is Galois of degree 4, so its Galois group has order 4. The elements of the Galois group are determined by their

### Galois theory of fields

1 Galois theory of fields This first chapter is both a concise introduction to Galois theory and a warmup for the more advanced theories to follow. We begin with a brisk but reasonably complete account

### Field Theory Problems

Field Theory Problems I. Degrees, etc. 1. Answer the following: (a Find u R such that Q(u = Q( 2, 3 5. (b Describe how you would find all w Q( 2, 3 5 such that Q(w = Q( 2, 3 5. 2. If a, b K are algebraic

### Polynomial Rings and Galois Theory

Polynomial Rings and Galois Theory Giacomo Micheli Balázs Szendrői Introduction Galois theory studies the symmetries of the roots of a polynomial equation. The existence of the two solutions (1) x 1,2

### Department of Mathematics, University of California, Berkeley

ALGORITHMIC GALOIS THEORY Hendrik W. Lenstra jr. Mathematisch Instituut, Universiteit Leiden Department of Mathematics, University of California, Berkeley K = field of characteristic zero, Ω = algebraically

### GALOIS GROUPS AS PERMUTATION GROUPS

GALOIS GROUPS AS PERMUTATION GROUPS KEITH CONRAD 1. Introduction A Galois group is a group of field automorphisms under composition. By looking at the effect of a Galois group on field generators we can

### MTH 401: Fields and Galois Theory

MTH 401: Fields and Galois Theory Semester 1, 2014-2015 Dr. Prahlad Vaidyanathan Contents Classical Algebra 3 I. Polynomials 6 1. Ring Theory.................................. 6 2. Polynomial Rings...............................

### Page Points Possible Points. Total 200

Instructions: 1. The point value of each exercise occurs adjacent to the problem. 2. No books or notes or calculators are allowed. Page Points Possible Points 2 20 3 20 4 18 5 18 6 24 7 18 8 24 9 20 10

### Course 311: Hilary Term 2006 Part IV: Introduction to Galois Theory

Course 311: Hilary Term 2006 Part IV: Introduction to Galois Theory D. R. Wilkins Copyright c David R. Wilkins 1997 2006 Contents 4 Introduction to Galois Theory 2 4.1 Polynomial Rings.........................

### Chapter 4. Fields and Galois Theory

Chapter 4 Fields and Galois Theory 63 64 CHAPTER 4. FIELDS AND GALOIS THEORY 4.1 Field Extensions 4.1.1 K[u] and K(u) Def. A field F is an extension field of a field K if F K. Obviously, F K = 1 F = 1

### AN INTRODUCTION TO GALOIS THEORY

AN INTRODUCTION TO GALOIS THEORY STEVEN DALE CUTKOSKY In these notes we consider the problem of constructing the roots of a polynomial. Suppose that F is a subfield of the complex numbers, and f(x) is

### List of topics for the preliminary exam in algebra

List of topics for the preliminary exam in algebra 1 Basic concepts 1. Binary relations. Reflexive, symmetric/antisymmetryc, and transitive relations. Order and equivalence relations. Equivalence classes.

### 9. Finite fields. 1. Uniqueness

9. Finite fields 9.1 Uniqueness 9.2 Frobenius automorphisms 9.3 Counting irreducibles 1. Uniqueness Among other things, the following result justifies speaking of the field with p n elements (for prime

### 22. Galois theory. G = Gal(L/k) = Aut(L/k) [L : K] = H. Gal(K/k) G/H

22. Galois theory 22.1 Field extensions, imbeddings, automorphisms 22.2 Separable field extensions 22.3 Primitive elements 22.4 Normal field extensions 22.5 The main theorem 22.6 Conjugates, trace, norm

### M345P11 Galois Theory

M345P11 Galois Theory Lectured by Prof Alessio Corti, notes taken by Wanlong Zheng Comments or corrections should be sent to wz3415@ic.ac.uk. Last updated: April 20, 2018 Contents 1 Introduction 2 1.1

### ALGEBRA 11: Galois theory

Galois extensions Exercise 11.1 (!). Consider a polynomial P (t) K[t] of degree n with coefficients in a field K that has n distinct roots in K. Prove that the ring K[t]/P of residues modulo P is isomorphic

### CONSTRUCTIBLE NUMBERS AND GALOIS THEORY

CONSTRUCTIBLE NUMBERS AND GALOIS THEORY SVANTE JANSON Abstract. We correct some errors in Grillet [2], Section V.9. 1. Introduction The purpose of this note is to correct some errors in Grillet [2], Section

### Finite Fields. [Parts from Chapter 16. Also applications of FTGT]

Finite Fields [Parts from Chapter 16. Also applications of FTGT] Lemma [Ch 16, 4.6] Assume F is a finite field. Then the multiplicative group F := F \ {0} is cyclic. Proof Recall from basic group theory

### Galois theory. Philippe H. Charmoy supervised by Prof Donna M. Testerman

Galois theory Philippe H. Charmoy supervised by Prof Donna M. Testerman Autumn semester 2008 Contents 0 Preliminaries 4 0.1 Soluble groups........................... 4 0.2 Field extensions...........................

### 1 Finite abelian groups

Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Each Problem is due one week from the date it is assigned. Do not hand them in early. Please put them on the desk in front of the room

### The Kummer Pairing. Alexander J. Barrios Purdue University. 12 September 2013

The Kummer Pairing Alexander J. Barrios Purdue University 12 September 2013 Preliminaries Theorem 1 (Artin. Let ψ 1, ψ 2,..., ψ n be distinct group homomorphisms from a group G into K, where K is a field.

### GALOIS THEORY AT WORK: CONCRETE EXAMPLES

GALOIS THEORY AT WORK: CONCRETE EXAMPLES KEITH CONRAD 1. Examples Example 1.1. The field extension Q(, 3)/Q is Galois of degree 4, so its Galois group has order 4. The elements of the Galois group are

### Some algebraic number theory and the reciprocity map

Some algebraic number theory and the reciprocity map Ervin Thiagalingam September 28, 2015 Motivation In Weinstein s paper, the main problem is to find a rule (reciprocity law) for when an irreducible

### ORAL QUALIFYING EXAM QUESTIONS. 1. Algebra

ORAL QUALIFYING EXAM QUESTIONS JOHN VOIGHT Below are some questions that I have asked on oral qualifying exams (starting in fall 2015). 1.1. Core questions. 1. Algebra (1) Let R be a noetherian (commutative)

### SOLVING SOLVABLE QUINTICS. D. S. Dummit

D. S. Dummit Abstract. Let f(x) = x 5 + px 3 + qx + rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if

### Notes on Galois Theory

Notes on Galois Theory Math 431 04/28/2009 Radford We outline the foundations of Galois theory. Most proofs are well beyond the scope of the our course and are therefore omitted. The symbols and in the

### ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

### Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition).

Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 14.2 Exercise 3. Determine the Galois group of (x 2 2)(x 2 3)(x 2 5). Determine all the subfields

### Galois Theory and Some Applications

Galois Theory and Some Applications Aparna Ramesh July 19, 2015 Introduction In this project, we study Galois theory and discuss some applications. The theory of equations and the ancient Greek problems

### MAIN THEOREM OF GALOIS THEORY

MAIN THEOREM OF GALOIS THEORY Theorem 1. [Main Theorem] Let L/K be a finite Galois extension. and (1) The group G = Gal(L/K) is a group of order [L : K]. (2) The maps defined by and f : {subgroups of G}!

### NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22

NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 RAVI VAKIL Hi Dragos The class is in 381-T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm

### By Dr. P.M.H. Wilson, L A TEXed by Matt Daws, Michaelmas Galois Theory, IIB

By Dr. P.M.H. Wilson, L A TEXed by Matt Daws, Michaelmas 1999 Email: matt.daws@cantab.net Galois Theory, IIB ii 0.1 Introduction These notes are based upon the lectures given by Dr. P.M.H. Wilson in Michaelmas

### Determining the Galois group of a rational polynomial

JAH 1 Determining the Galois group of a rational polynomial Alexander Hulpke Department of Mathematics Colorado State University Fort Collins, CO, 80523 hulpke@math.colostate.edu http://www.math.colostate.edu/

### Fields and Galois Theory Fall 2004 Professor Yu-Ru Liu

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............................................

### Galois Theory of Cyclotomic Extensions

Galois Theory of Cyclotomic Extensions Winter School 2014, IISER Bhopal Romie Banerjee, Prahlad Vaidyanathan I. Introduction 1. Course Description The goal of the course is to provide an introduction to

### ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008

ALGEBRA PH.D. QUALIFYING EXAM September 27, 2008 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved completely

### MAT 535 Problem Set 5 Solutions

Final Exam, Tues 5/11, :15pm-4:45pm Spring 010 MAT 535 Problem Set 5 Solutions Selected Problems (1) Exercise 9, p 617 Determine the Galois group of the splitting field E over F = Q of the polynomial f(x)

### GALOIS THEORY I (Supplement to Chapter 4)

GALOIS THEORY I (Supplement to Chapter 4) 1 Automorphisms of Fields Lemma 1 Let F be a eld. The set of automorphisms of F; Aut (F ) ; forms a group (under composition of functions). De nition 2 Let F be

### x by so in other words ( )

Math 210B. Norm and trace An interesting application of Galois theory is to help us understand properties of two special constructions associated to field extensions, the norm and trace. If L/k is a finite

### Galois Theory. This material is review from Linear Algebra but we include it for completeness.

Galois Theory Galois Theory has its origins in the study of polynomial equations and their solutions. What is has revealed is a deep connection between the theory of fields and that of groups. We first

### Section V.6. Separability

V.6. Separability 1 Section V.6. Separability Note. Recall that in Definition V.3.10, an extension field F is a separable extension of K if every element of F is algebraic over K and every root of the

Algebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.

### Section V.7. Cyclic Extensions

V.7. Cyclic Extensions 1 Section V.7. Cyclic Extensions Note. In the last three sections of this chapter we consider specific types of Galois groups of Galois extensions and then study the properties of

### MT5836 Galois Theory MRQ

MT5836 Galois Theory MRQ May 3, 2017 Contents Introduction 3 Structure of the lecture course............................... 4 Recommended texts..................................... 4 1 Rings, Fields and

### A Field Extension as a Vector Space

Chapter 8 A Field Extension as a Vector Space In this chapter, we take a closer look at a finite extension from the point of view that is a vector space over. It is clear, for instance, that any is a linear

### Fields and Galois Theory

Fields and Galois Theory Rachel Epstein September 12, 2006 All proofs are omitted here. They may be found in Fraleigh s A First Course in Abstract Algebra as well as many other algebra and Galois theory

### Polynomial Rings and Galois Theory

Polynomial Rings and Galois Theory Balázs Szendrői Introduction Galois theory studies the symmetries of the roots of a polynomial equation. The existence of the two solutions (1) x 1,2 = b ± p b 2 4c 2

### Algebra Questions. May 13, Groups 1. 2 Classification of Finite Groups 4. 3 Fields and Galois Theory 5. 4 Normal Forms 9

Algebra Questions May 13, 2013 Contents 1 Groups 1 2 Classification of Finite Groups 4 3 Fields and Galois Theory 5 4 Normal Forms 9 5 Matrices and Linear Algebra 10 6 Rings 11 7 Modules 13 8 Representation

### NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

### Module MA3411: Abstract Algebra Galois Theory Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents 1 Basic Principles of Group Theory 1 1.1 Groups...............................

### AN ALTERNATIVE APPROACH TO THE CONCEPT OF SEPARABILITY IN GALOIS THEORY arxiv: v1 [math.ac] 27 Sep 2017

AN ALTERNATIVE APPROACH TO THE CONCEPT OF SEPARABILITY IN GALOIS THEORY arxiv:1709.09640v1 [math.ac] 27 Sep 2017 M. G. MAHMOUDI Abstract. The notion of a separable extension is an important concept in

Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.

### NOTES ON GALOIS THEORY Alfonso Gracia-Saz, MAT 347

NOTES ON GALOIS THEORY Alfonso Gracia-Saz, MAT 347 Go to the roots of these calculations! Group the operations. Classify them according to their complexities rather than their appearances! This, I believe,

### THE GALOIS CORRESPONDENCE

THE GALOIS CORRESPONDENCE KEITH CONRAD 1. Introduction Let L/K be a field extension. A K-automorphism of L is a field automorphism σ : L L which fixes the elements of K: σ(c) = c for all c K. The set of

### THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - SPRING SESSION ADVANCED ALGEBRA II.

THE JOHNS HOPKINS UNIVERSITY Faculty of Arts and Sciences FINAL EXAM - SPRING SESSION 2006 110.402 - ADVANCED ALGEBRA II. Examiner: Professor C. Consani Duration: 3 HOURS (9am-12:00pm), May 15, 2006. No

### Section V.8. Cyclotomic Extensions

V.8. Cyclotomic Extensions 1 Section V.8. Cyclotomic Extensions Note. In this section we explore splitting fields of x n 1. The splitting fields turn out to be abelian extensions (that is, algebraic Galois

### A BRIEF INTRODUCTION TO LOCAL FIELDS

A BRIEF INTRODUCTION TO LOCAL FIELDS TOM WESTON The purpose of these notes is to give a survey of the basic Galois theory of local fields and number fields. We cover much of the same material as [2, Chapters

### 22M: 121 Final Exam. Answer any three in this section. Each question is worth 10 points.

22M: 121 Final Exam This is 2 hour exam. Begin each question on a new sheet of paper. All notations are standard and the ones used in class. Please write clearly and provide all details of your work. Good

### TC10 / 3. Finite fields S. Xambó

TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the

### Keywords and phrases: Fundamental theorem of algebra, constructible

Lecture 16 : Applications and Illustrations of the FTGT Objectives (1) Fundamental theorem of algebra via FTGT. (2) Gauss criterion for constructible regular polygons. (3) Symmetric rational functions.

### Q N id β. 2. Let I and J be ideals in a commutative ring A. Give a simple description of

Additional Problems 1. Let A be a commutative ring and let 0 M α N β P 0 be a short exact sequence of A-modules. Let Q be an A-module. i) Show that the naturally induced sequence is exact, but that 0 Hom(P,

### Galois Theory and the Insolvability of the Quintic Equation

Galois Theory and the Insolvability of the Quintic Equation Daniel Franz 1. Introduction Polynomial equations and their solutions have long fascinated mathematicians. The solution to the general quadratic

### Notes on Field Extensions

Notes on Field Extensions Ryan C. Reich 16 June 2006 1 Definitions Throughout, F K is a finite field extension. We fix once and for all an algebraic closure M for both and an embedding of F in M. When

### Math 121. Fundamental Theorem and an example

Math 121. Fundamental Theorem and an example Let K/k be a finite Galois extension and G = Gal(K/k), so #G = [K : k] by the counting criterion for separability discussed in class. In this handout we will

### Math 121 Homework 6 Solutions

Math 11 Homework 6 Solutions Problem 14. # 17. Let K/F be any finite extension and let α K. Let L be a Galois extension of F containing K and let H Gal(L/F ) be the subgroup corresponding to K. Define

### Math 414 Answers for Homework 7

Math 414 Answers for Homework 7 1. Suppose that K is a field of characteristic zero, and p(x) K[x] an irreducible polynomial of degree d over K. Let α 1, α,..., α d be the roots of p(x), and L = K(α 1,...,α

### disc f R 3 (X) in K[X] G f in K irreducible S 4 = in K irreducible A 4 in K reducible D 4 or Z/4Z = in K reducible V Table 1

GALOIS GROUPS OF CUBICS AND QUARTICS IN ALL CHARACTERISTICS KEITH CONRAD 1. Introduction Treatments of Galois groups of cubic and quartic polynomials usually avoid fields of characteristic 2. Here we will

### 1. Group Theory Permutations.

1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7

### Notes on graduate algebra. Robert Harron

Notes on graduate algebra Robert Harron Department of Mathematics, Keller Hall, University of Hawai i at Mānoa, Honolulu, HI 96822, USA E-mail address: rharron@math.hawaii.edu Abstract. Graduate algebra

### IUPUI Qualifying Exam Abstract Algebra

IUPUI Qualifying Exam Abstract Algebra January 2017 Daniel Ramras (1) a) Prove that if G is a group of order 2 2 5 2 11, then G contains either a normal subgroup of order 11, or a normal subgroup of order

### Section 33 Finite fields

Section 33 Finite fields Instructor: Yifan Yang Spring 2007 Review Corollary (23.6) Let G be a finite subgroup of the multiplicative group of nonzero elements in a field F, then G is cyclic. Theorem (27.19)