1. Group Theory Permutations.


 Johnathan Blake
 6 years ago
 Views:
Transcription
1 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 be of maximal order. What is π? Is π an even permutation? If ρ = π, is ρ conjugate to π? Problem 1.3. Let n 3, and let σ S n be an (n 1)cycle. Show that C Sn (σ) = σ. Problem 1.4. Let σ = (123)(456) S 6. Find the size of the conjugacy class of σ, and the order of the centralizer of σ in S 6. Problem 1.5. S n has an element of order 2(n 2) whenever n > 1 is odd Sylow theorems. Problem 1.6. How many elements of order 7 are there in a simple group of order 168? Problem 1.7. A group of order 48 must have a normal subgroup of order 8 or 16. Problem 1.8. Let G be a group of order p a b where (p, b) = 1, b > 1 and p a b > b!. Then G is not simple. In particular, there is no simple group of order 36. Problem 1.9. Let G be a finite group, H G, and let P be a Sylow psubgroup of H. Assume that N G (P ) H and conclude that P is a Sylow psubgroup of G. Problem A group of order 340 has a normal cyclic subgroup of order 85 and an abelian subgroup of order 4. Problem A group of order 108 has a normal subgroup of order 9 or 27. Problem Let p, q be distinct primes. Then any group of order p 2 q either has a normal Sylow psubgroup or it has a normal Sylow qsubgroup. Problem Proof Frattini argument, i.e.: Let G be a finite group, H G, P a Sylow psubgroup of H. Then G = N G (P )H. Problem Let N be a normal subgroup of G that contains a Sylow psubgroup of G. Then the number of Sylow psubgroups of N is the same as that of G. Problem Let G be a group of order 105, then G has a normal Sylow 5subgroup and a normal Sylow 7subgroup Actions. Problem Let G be a finite simple group containing an element of order 21. Show that every proper subgroup of G is of index at least 10. Problem Let G be a simple group with G > 60. Show that G has no subgroups of index less or equal to 5. Problem Let G be a group acting on set S, H G, such that the inherited action of H on S is transitive. Show that for every t S we have G = HG t, where G t is the stabilizer of t. Problem Let G act on Σ transitively, α, β Σ, α β. Show that G α G β G. Problem Show that S 6 has no simple subgroup of order 180 (index 4). Then use this fact to prove that no simple group of order 180 = exists. Problem Prove that there are no simple groups of order 1452 =
2 Solvability and nilpotency. Problem It is true (and you can take it for granted) that a group of order 12 is either isomorphic to A 4 or contains an element of order 6. Show that a group of order 12p is solvable whenever p > 11 is a prime. Problem Show that every group of order is solvable. Problem Let G be a group of order 780 = that is not solvable. What are the composition factors of G? You can take for granted that the only nonabelian simple group of order at most 60 is A Classical problems. Problem Show that the additive group Q of the rational numbers is not cyclic. Show that every finitely generated subgroup of Q is cyclic Other structural problems. Problem Let G be a simple group of order at least 3. Then Z(Aut(G)) = 1 if and only if G is not abelian. Problem Let M be a maximal subgroup of a finite group G. If M G, show that [G : M] is prime. Problem Let G be a nonabelian group of order p 3. Show that Z(G) = G is a subgroup of order p, and that G/Z(G) = Z p Z p. Problem A finite group whose only automorphism is the identity map must have order at most two. Problem Let G be a finite group, N G, H G, ([G : N], H ) = 1. Then H N. Problem Find all finite groups that have exactly two conjugacy classes. Problem Find all abelian groups of order 108 (up to isomorphism) Ideals. 2. Rings Problem 2.1. Let R be a commutative ring with 1. Show that the sum of any two principal ideals of R is principal if and only if every finitely generated ideal of R is principal. Problem 2.2. Let R 1, R 2 be commutative rings with identities and let R = R 1 R 2. Show that every ideal I of R is of the form I = I 1 I 2 with I i an ideal of R i, for i = 1, 2. Problem 2.3. Let R be a commutative ring with unity, S a subset of R closed under multiplication, and P a maximal element of the set {I; I is an ideal of R such that I S = }. Show that P is a prime ideal of R. Problem 2.4. Let R be a commutative ring with 1. Show that if M is a maximal ideal of R then M is a prime ideal of R. Problem 2.5. Let R be a commutative ring that is not a field, and let P 0 be a maximal ideal of R. Show that P [x] is a prime ideal of R[x] but P [x] is not a maximal ideal of R[x].
3 Problem 2.6. Give an example of a commutative ring R with prime ideal P 0 that is not maximal. Problem 2.7. Let R be a commutative ring with 1 0 such that every ideal of R is prime. Then R is a field. Problem 2.8. Let R be an integral domain with 1 0 where IJ = I J for all ideals I, J of R. Then R is a field. Problem 2.9. Let R be a commutative ring with 1. We say that R satisfies the ACC if whenever I 1 I 2 I 3 is a chain of ideals of R, then there exists an integer N such that I k = I N for every k N. Prove that R satisfies the ACC if and only if every ideal of R is finitely generated. Problem Let F be a field and F [x, y] the ring of polynomials in variables x, y. Show that the ideal of F [x, y] generated by {x, y} is not principal. Problem Let R be a commutative ring with 1 such that it has precisely three ideals, say {0}, I and R. Then (i) every a R \ I is a unit, (ii) if a, b I then ab = 0. Problem Let R be the ring of 2 2 matrices with coefficients in some field F. Show that R is simple, i.e., it has no proper nonzero ideals. Problem Let C be a chain of prime ideals of a commutative ring R with 1. (Do not assume that the chain has a minimal or maximal element, nor that it is finite, countable, etc.) Show that C C C and C C C are prime ideals of R. Problem Let p be a prime and R the ring of all 2 2 matrices of the form ( a b pb a), where a, b Z. Then R is isomorphic to Z[ p]. Problem Let R be a nonzero commutative ring with 1. Show that if I is an ideal of R such that 1 + a is a unit for every a I, then I is contained in every maximal ideal of R. Problem Let R be a commutative ring with 1, and let P be a prime ideal of R. Show that there is a prime ideal P 0 P that contains no prime ideals properly. Problem A commutative ring with 1 is local if it has a unique maximal ideal. Let R be a commutative ring with 1. Show that R is local iff the set of nonunits in R is an ideal Irreducible and prime elements. Problem Prove that in an integral domain R every prime element is irreducible. Problem Construct the following: (i) An integral domain with an irreducible element that is not prime. (ii) A commutative ring with a prime element that is not irreducible. Problem Let D be a UFD and F the field of fractions of D. irreducible in D. Prove that there is no x F such that x 2 = d. 3 Let d D be Problem Let D be an integral domain an c an irreducible element of D. Show that the ideal (x, c) is not a principal ideal in D[x]. Then show that the same conclusion cannot be reached if the assumption that c is irreducible is dropped.
4 Polynomials and quotient rings. Problem Find all values of a Z 5 such that the quotient ring R = Z 5 [x]/(x 3 + 2x 2 + ax + 3) is a field. Problem Show that the ring R = Z 2 [x]/(x 2 + 1) has 4 elements and that it is not isomorphic to Z 4 nor Z 2 Z UFDs. Problem Let D = Z[ 21] = {m + n 21; m, n Z}, and let F = Q( 21) be the field of fractions of D. Show that: (i) x 2 x 5 is irreducible in D[x] but not in F [x], (ii) D is not a UFD, (iii) D[x] is not a UFD. Problem Let Z[1/2] = {a/2 n ; ; a, n Z, n 0}. Show that Z[x]/(2x 1) = Z[1/2]. Then find an ideal I of Z[x] such that (2x 1) I Z[x] Examples. The following problem would not appear in its entirety as a single question on the qualifier. Problem Give an example for each of the following: (i) An infinite noncommutative ring with nonzero characteristic. (ii) An integral domain which is not a unique factorization domain. (iii) A noncommutative domain that is not a division ring. (iv) A nonzero prime ideal of a commutative ring that is not a maximal ideal. (v) A commutative ring with a sequence of prime ideals {P n } n=1 such that P i P i+1. (vi) A commutative ring that has exactly one maximal ideal and is not a field. (vii) A finite noncommutative ring. (viii) A noncommutative ring with exactly two maximal ideals. (ix) An infinite noncommutative ring with only finitely many ideals. (x) A unique factorization domain that is not a principal ideal domain. (xi) An infinite domain of nonzero characteristic Classical problems. Problem Let R be the ring of 2 2 real matrices of the form ( a b b a). Show that R is isomorphic to C. Proof. Let R be a nonzero ring with 1. Show that every proper ideal of R is contained in a maximal ideal. Problem Let R = Z/60Z. How many units does R have? How many ideals does R have? Problem Let R be a commutative ring with identity. Show that the nilpotent elements form an ideal of R Finite Fields. 3. Fields Problem 3.1. Construct a field of order 8. Problem 3.2. Construct a field of order 9.
5 Problem 3.3. Let α be a root of x in an extension of Z 3. Let F = Z 3 (α) and f(x) = x Z 3 [x]. Then: (i) Show that f splits in K. (ii) Find a generator β of K. (iii) Express the roots of f in terms of β Extensions. Problem 3.4. Let D be an integral domain with subring F that happens to be a field. Assume that D is algebraic over F. Then D is a field. Problem 3.5. Show that every finite extension of fields is algebraic. Problem 3.6. Let F E. Show that the algebraic closure of F in E defined by A = {u E; u is algebraic over F } is a field. Problem 3.7. Give an example of an algebraic extension that is not finite. Problem 3.8. Find the minimal polynomial of u = over Q Splitting Fields. Problem 3.9. Let E be a spitting field of f(x) F [x] over F, where deg f = n. Show that E : F n!. Problem Find a splitting field E of f(x) = x 4 2x 2 3 over Q. Assume that E is another splitting field of f over Q. What is E : Q? Problem Let E be a splitting field of f(x) = x 3 5 over Q. What is E : Q? Problem Let E be a splitting field of f F [x] over F, where deg f = 2. Show that F E is a simple extension. Problem Let A be the splitting field of f(x) = x 2 2 over Q, and B the splitting field of g(x) = x 2 2x 1 over Q. Show that A = B Theoretical problems. 4. Galois Theory Problem 4.1. Define: Galois group, Galois extension, Fixed field. Problem 4.2. Show that every finite extension of a finite field is a Galois extension Computational problems. Problem 4.3. In each case show that F E is Galois, determine the Galois group (isomorphism type and elements), find the lattice of intermediate fields, and describe the intermediate fields. (i) E = Q(u), u = e 2pi/5, F = Q, (ii) E = Q(i, 3), F = Q, (iii) E = Z 2 (u), where u is a root of x 4 + x + 1, F = Z 2. Problem 4.4. Determine the Galois group of x 4 1 and x 12 1 over Z 2. 5
QUALIFYING EXAM IN ALGEBRA August 2011
QUALIFYING EXAM IN ALGEBRA August 2011 1. There are 18 problems on the exam. Work and turn in 10 problems, in the following categories. I. Linear Algebra 1 problem II. Group Theory 3 problems III. Ring
More informationCSIR  Algebra Problems
CSIR  Algebra Problems N. Annamalai DST  INSPIRE Fellow (SRF) Department of Mathematics Bharathidasan University Tiruchirappalli 620024 Email: algebra.annamalai@gmail.com Website: https://annamalaimaths.wordpress.com
More informationPh.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018
Ph.D. Qualifying Examination in Algebra Department of Mathematics University of Louisville January 2018 Do 6 problems with at least 2 in each section. Group theory problems: (1) Suppose G is a group. The
More informationSample algebra qualifying exam
Sample algebra qualifying exam University of Hawai i at Mānoa Spring 2016 2 Part I 1. Group theory In this section, D n and C n denote, respectively, the symmetry group of the regular ngon (of order 2n)
More informationALGEBRA QUALIFYING EXAM SPRING 2012
ALGEBRA QUALIFYING EXAM SPRING 2012 Work all of the problems. Justify the statements in your solutions by reference to specific results, as appropriate. Partial credit is awarded for partial solutions.
More informationAlgebra 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.
More informationGroup Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.
Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite
More informationA PROOF OF BURNSIDE S p a q b THEOREM
A PROOF OF BURNSIDE S p a q b THEOREM OBOB Abstract. We prove that if p and q are prime, then any group of order p a q b is solvable. Throughout this note, denote by A the set of algebraic numbers. We
More informationφ(xy) = (xy) n = x n y n = φ(x)φ(y)
Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =
More informationIUPUI 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
More informationAlgebra Qualifying Exam, Fall 2018
Algebra Qualifying Exam, Fall 2018 Name: Student ID: Instructions: Show all work clearly and in order. Use full sentences in your proofs and solutions. All answers count. In this exam, you may use the
More informationALGEBRA 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
More informationAlgebra Exam, Spring 2017
Algebra Exam, Spring 2017 There are 5 problems, some with several parts. Easier parts count for less than harder ones, but each part counts. Each part may be assumed in later parts and problems. Unjustified
More informationAlgebra SEP Solutions
Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since
More informationAlgebra Homework, Edition 2 9 September 2010
Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.
More informationPROBLEMS FROM GROUP THEORY
PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.
More informationAlgebra Exam Topics. Updated August 2017
Algebra Exam Topics Updated August 2017 Starting Fall 2017, the Masters Algebra Exam will have 14 questions. Of these students will answer the first 8 questions from Topics 1, 2, and 3. They then have
More informationPage 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
More informationGraduate Preliminary Examination
Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counterexample to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.
More informationMath 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
More informationGroup Theory. Ring and Module Theory
Department of Mathematics University of South Florida QUALIFYING EXAM ON ALGEBRA Saturday, May 14, 016 from 9:00 am to 1:00 noon Examiners: Brian Curtin and Dmytro Savchuk This is a three hour examination.
More informationGalois 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
More informationTEST CODE: PMB SYLLABUS
TEST CODE: PMB SYLLABUS Convergence and divergence of sequence and series; Cauchy sequence and completeness; BolzanoWeierstrass theorem; continuity, uniform continuity, differentiability; directional
More informationSUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT
SUMMARY ALGEBRA I LOUISPHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.
More informationAlgebra Ph.D. Entrance Exam Fall 2009 September 3, 2009
Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Directions: Solve 10 of the following problems. Mark which of the problems are to be graded. Without clear indication which problems are to be graded
More informationProblem 4 (Wed Jan 29) Let G be a finite abelian group. Prove that the following are equivalent
Last revised: May 16, 2014 A.Miller M542 www.math.wisc.edu/ miller/ Problem 1 (Fri Jan 24) (a) Find an integer x such that x = 6 mod 10 and x = 15 mod 21 and 0 x 210. (b) Find the smallest positive integer
More informationMath 553 Qualifying Exam. In this test, you may assume all theorems proved in the lectures. All other claims must be proved.
Math 553 Qualifying Exam January, 2019 Ron Ji In this test, you may assume all theorems proved in the lectures. All other claims must be proved. 1. Let G be a group of order 3825 = 5 2 3 2 17. Show that
More informationALGEBRA 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)
More informationAlgebra Qualifying Exam Solutions. Thomas Goller
Algebra Qualifying Exam Solutions Thomas Goller September 4, 2 Contents Spring 2 2 2 Fall 2 8 3 Spring 2 3 4 Fall 29 7 5 Spring 29 2 6 Fall 28 25 Chapter Spring 2. The claim as stated is false. The identity
More informationExercises on chapter 1
Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G
More informationKevin James. pgroups, Nilpotent groups and Solvable groups
pgroups, Nilpotent groups and Solvable groups Definition A maximal subgroup of a group G is a proper subgroup M G such that there are no subgroups H with M < H < G. Definition A maximal subgroup of a
More information1 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
More informationReal Analysis Prelim Questions Day 1 August 27, 2013
Real Analysis Prelim Questions Day 1 August 27, 2013 are 5 questions. TIME LIMIT: 3 hours Instructions: Measure and measurable refer to Lebesgue measure µ n on R n, and M(R n ) is the collection of measurable
More informationFields 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
More information1 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
More informationAlgebra Exercises in group theory
Algebra 3 2010 Exercises in group theory February 2010 Exercise 1*: Discuss the Exercises in the sections 1.11.3 in Chapter I of the notes. Exercise 2: Show that an infinite group G has to contain a nontrivial
More information6]. (10) (i) Determine the units in the rings Z[i] and Z[ 10]. If n is a squarefree
Quadratic extensions Definition: Let R, S be commutative rings, R S. An extension of rings R S is said to be quadratic there is α S \R and monic polynomial f(x) R[x] of degree such that f(α) = 0 and S
More informationSchool 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
More informationAlgebra 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
More informationList 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.
More informationMay 6, Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work.
Math 236H May 6, 2008 Be sure to write your name on your bluebook. Use a separate page (or pages) for each problem. Show all of your work. 1. (15 points) Prove that the symmetric group S 4 is generated
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings These notes are a summary of some of the important points on divisibility in polynomial rings from 17 and 18. PIDs Definition 1 A principal ideal domain (PID) is an integral
More informationALGEBRA QUALIFYING EXAM PROBLEMS
ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General
More informationAlgebra Prelim Notes
Algebra Prelim Notes Eric Staron Summer 2007 1 Groups Define C G (A) = {g G gag 1 = a for all a A} to be the centralizer of A in G. In particular, this is the subset of G which commuted with every element
More informationMath 120 HW 9 Solutions
Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z
More informationALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011
ALGEBRA PH.D. QUALIFYING EXAM SOLUTIONS October 20, 2011 A passing paper consists of four problems solved completely plus significant progress on two other problems; moreover, the set of problems solved
More informationHomework problems from Chapters IVVI: answers and solutions
Homework problems from Chapters IVVI: answers and solutions IV.21.1. In this problem we have to describe the field F of quotients of the domain D. Note that by definition, F is the set of equivalence
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. 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] All of
More informationPRACTICE FINAL MATH , MIT, SPRING 13. You have three hours. This test is closed book, closed notes, no calculators.
PRACTICE FINAL MATH 18.703, MIT, SPRING 13 You have three hours. This test is closed book, closed notes, no calculators. There are 11 problems, and the total number of points is 180. Show all your work.
More informationRings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.
Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary
More informationMath 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours
Math 120. Groups and Rings Midterm Exam (November 8, 2017) 2 Hours Name: Please read the questions carefully. You will not be given partial credit on the basis of having misunderstood a question, and please
More informationAlgebraic Structures Exam File Fall 2013 Exam #1
Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x 4 + 16 = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write
More informationAlgebraic structures I
MTH5100 Assignment 110 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one
More informationOhio State University Department of Mathematics Algebra Qualifier Exam Solutions. Timothy All Michael Belfanti
Ohio State University Department of Mathematics Algebra Qualifier Exam Solutions Timothy All Michael Belfanti July 22, 2013 Contents Spring 2012 1 1. Let G be a finite group and H a nonnormal subgroup
More informationGALOIS 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
More informationGalois 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
More informationCHAPTER 14. Ideals and Factor Rings
CHAPTER 14 Ideals and Factor Rings Ideals Definition (Ideal). A subring A of a ring R is called a (twosided) ideal of R if for every r 2 R and every a 2 A, ra 2 A and ar 2 A. Note. (1) A absorbs elements
More informationSUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III Week 1 Lecture 1 Tuesday 3 March.
SUMMARY OF GROUPS AND RINGS GROUPS AND RINGS III 2009 Week 1 Lecture 1 Tuesday 3 March. 1. Introduction (Background from Algebra II) 1.1. Groups and Subgroups. Definition 1.1. A binary operation on a set
More informationPart IX. Factorization
IX.45. Unique Factorization Domains 1 Part IX. Factorization Section IX.45. Unique Factorization Domains Note. In this section we return to integral domains and concern ourselves with factoring (with respect
More information2a 2 4ac), provided there is an element r in our
MTH 310002 Test II Review Spring 2012 Absractions versus examples The purpose of abstraction is to reduce ideas to their essentials, uncluttered by the details of a specific situation Our lectures built
More informationMath 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).
More informationSupplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.
Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is
More informationNONNILPOTENT GROUPS WITH THREE CONJUGACY CLASSES OF NONNORMAL SUBGROUPS. Communicated by Alireza Abdollahi. 1. Introduction
International Journal of Group Theory ISSN (print): 22517650, ISSN (online): 22517669 Vol. 3 No. 2 (2014), pp. 17. c 2014 University of Isfahan www.theoryofgroups.ir www.ui.ac.ir NONNILPOTENT GROUPS
More informationMATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:0010:00 PM
MATH 3030, Abstract Algebra FALL 2012 Toby Kenney Midyear Examination Friday 7th December: 7:0010:00 PM Basic Questions 1. Compute the factor group Z 3 Z 9 / (1, 6). The subgroup generated by (1, 6) is
More informationAlgebra Ph.D. Preliminary Exam
RETURN THIS COVER SHEET WITH YOUR EXAM AND SOLUTIONS! Algebra Ph.D. Preliminary Exam August 18, 2008 INSTRUCTIONS: 1. Answer each question on a separate page. Turn in a page for each problem even if you
More informationChapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples
Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter
More informationAlgebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3...
Algebra Exam Fall 2006 Alexander J. Wertheim Last Updated: October 26, 2017 Contents 1 Groups 2 1.1 Problem 1..................................... 2 1.2 Problem 2..................................... 2
More informationMAT 535 Problem Set 5 Solutions
Final Exam, Tues 5/11, :15pm4:45pm 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)
More informationJanuary 2016 Qualifying Examination
January 2016 Qualifying Examination If you have any difficulty with the wording of the following problems please contact the supervisor immediately. All persons responsible for these problems, in principle,
More information1 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
More informationThe 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
More informationTotal 100
Math 542 Midterm Exam, Spring 2016 Prof: Paul Terwilliger Your Name (please print) SOLUTIONS NO CALCULATORS/ELECTRONIC DEVICES ALLOWED. MAKE SURE YOUR CELL PHONE IS OFF. Problem Value 1 10 2 10 3 10 4
More information(Rgs) Rings Math 683L (Summer 2003)
(Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that
More informationMATH 3030, Abstract Algebra Winter 2012 Toby Kenney Sample Midterm Examination Model Solutions
MATH 3030, Abstract Algebra Winter 2012 Toby Kenney Sample Midterm Examination Model Solutions Basic Questions 1. Give an example of a prime ideal which is not maximal. In the ring Z Z, the ideal {(0,
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationA. (Groups of order 8.) (a) Which of the five groups G (as specified in the question) have the following property: G has a normal subgroup N such that
MATH 402A  Solutions for the suggested problems. A. (Groups of order 8. (a Which of the five groups G (as specified in the question have the following property: G has a normal subgroup N such that N =
More informationRINGS: SUMMARY OF MATERIAL
RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 1113 of Artin. Definitions not included here may be considered
More informationModern Computer Algebra
Modern Computer Algebra Exercises to Chapter 25: Fundamental concepts 11 May 1999 JOACHIM VON ZUR GATHEN and JÜRGEN GERHARD Universität Paderborn 25.1 Show that any subgroup of a group G contains the neutral
More informationSection 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)
More informationAlgebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001
Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,
More information(3) Let Y be a totally bounded subset of a metric space X. Then the closure Y of Y
() Consider A = { q Q : q 2 2} as a subset of the metric space (Q, d), where d(x, y) = x y. Then A is A) closed but not open in Q B) open but not closed in Q C) neither open nor closed in Q D) both open
More informationSelected 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
More informationAugust 2015 Qualifying Examination Solutions
August 2015 Qualifying Examination Solutions If you have any difficulty with the wording of the following problems please contact the supervisor immediately. All persons responsible for these problems,
More informationSelected 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 2.1 Exercise (6). Let G be an abelian group. Prove that T = {g G g < } is a subgroup of G.
More informationGalois 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
More informationYale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall Midterm Exam Review Solutions
Yale University Department of Mathematics Math 350 Introduction to Abstract Algebra Fall 2015 Midterm Exam Review Solutions Practice exam questions: 1. Let V 1 R 2 be the subset of all vectors whose slope
More information38 Irreducibility criteria in rings of polynomials
38 Irreducibility criteria in rings of polynomials 38.1 Theorem. Let p(x), q(x) R[x] be polynomials such that p(x) = a 0 + a 1 x +... + a n x n, q(x) = b 0 + b 1 x +... + b m x m and a n, b m 0. If b m
More information2 (17) Find nontrivial left and right ideals of the ring of 22 matrices over R. Show that there are no nontrivial two sided ideals. (18) State and pr
MATHEMATICS Introduction to Modern Algebra II Review. (1) Give an example of a noncommutative ring; a ring without unit; a division ring which is not a eld and a ring which is not a domain. (2) Show that
More informationA connection between number theory and linear algebra
A connection between number theory and linear algebra Mark Steinberger Contents 1. Some basics 1 2. Rational canonical form 2 3. Prime factorization in F[x] 4 4. Units and order 5 5. Finite fields 7 6.
More informationAlgebra. Travis Dirle. December 4, 2016
Abstract Algebra 2 Algebra Travis Dirle December 4, 2016 2 Contents 1 Groups 1 1.1 Semigroups, Monoids and Groups................ 1 1.2 Homomorphisms and Subgroups................. 2 1.3 Cyclic Groups...........................
More informationSPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS
SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly
More informationSolutions of exercise sheet 8
DMATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 8 1. In this exercise, we will give a characterization for solvable groups using commutator subgroups. See last semester s (Algebra
More informationSection III.6. Factorization in Polynomial Rings
III.6. Factorization in Polynomial Rings 1 Section III.6. Factorization in Polynomial Rings Note. We push several of the results in Section III.3 (such as divisibility, irreducibility, and unique factorization)
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,
More informationCHARACTER THEORY OF FINITE GROUPS. Chapter 1: REPRESENTATIONS
CHARACTER THEORY OF FINITE GROUPS Chapter 1: REPRESENTATIONS G is a finite group and K is a field. A Krepresentation of G is a homomorphism X : G! GL(n, K), where GL(n, K) is the group of invertible n
More informationA few exercises. 1. Show that f(x) = x 4 x 2 +1 is irreducible in Q[x]. Find its irreducible factorization in
A few exercises 1. Show that f(x) = x 4 x 2 +1 is irreducible in Q[x]. Find its irreducible factorization in F 2 [x]. solution. Since f(x) is a primitive polynomial in Z[x], by Gauss lemma it is enough
More information9. 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
More informationAbstract Algebra, Second Edition, by John A. Beachy and William D. Blair. Corrections and clarifications
1 Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair Corrections and clarifications Note: Some corrections were made after the first printing of the text. page 9, line 8 For of the
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 1. Let R 0 be a commutative ring with 1 and let S R be the subset of nonzero elements which are not zero divisors. (a)
More informationQ 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 Amodules. Let Q be an Amodule. i) Show that the naturally induced sequence is exact, but that 0 Hom(P,
More information