May 19, 2008 Math 220C Problem Set #6 Due: Wednesday, May 28 Grove, Chapter V: Exercise 1.2 on p. 172 Exercise 1.3 on p. 174 Exercise 1.4 on p. 176 Exercises 2.1 and 2.2 on p. 177 Exercise 3.1 on p. 182
Problem Set #5 Solutions Chapter VI.8. 8.1.1. Write out the permutation matrix representation of the dihedral group D4 corresponding to its action on the vertices of a square. Let S = {1, 2, 3, 4} be the vertices of a square. In terms of generators a
April 21, 2008 Math 220C Problem Set #3 Due: Wednesday, April 30 Grove, Chapter III: Exercise 4.1 on p. 96 Exercise 4.2 on p. 97 Exercise 4.3 on p. 98 Exercise 4.4 on p. 99 Exercise 4.5 on p. 101 Further Exercise 11 on p. 119 Further Exercise
April 30, 2008 Math 220C Problem Set #4 Due: Friday, May 9 Grove, Chapter III: Exercise 5.1 and 5.2 on p. 103 Exercise 5.3 on p. 105 Exercise 6.1 on p. 111 Exercise 6.2 on p. 113 Furthers Exercise 34 and 37 on p. 122 Further Exercise 48 on p. 1
Problem Set #3 Solutions Chapter III. 4.1 If f (x) F [x] and K is a splitting field for f (x) over F , denote by S the set of distinct roots of f (x) in K and let G = G(K : F ). i. If f (x) is irreducible over F , then G is transitive on S. Since K
Math 220C Galois theory of x3 - 2
April 11, 2008
We want to construct the splitting field for x3 - 2 over Q, and explicitly find its Galois group and the intermediate subfields. Since x3 - 2 is irreducible in Q[x], adjoining a root will give a fiel
Math 220C Further examples of group representations
May 9, 2008
In class, we discussed several additional examples of group representations. (1) Present Zn as x | xn = 1 , and let C be a primitive nth root of unity. ^ Consider the matrix represen
Problem Set #1 Solutions Chapter II. 1.2. Let A denote the algebraic numbers, i.e., the set of a C which are algebraic over Q. Then [A : Q] = . Proof. Suppose to the contrary that [A : Q] = n. Let a = 21/n+1 be a root of f (x) = xn+1 - 2 Q[x]; then
March 31, 2008 Math 220C Problem Set #1 Due: Monday, April 7 Grove, Chapter III: Exercise 1.1 on p. 82 Exercise 1.2 on p. 83 Exercise 1.4 on p. 85 Exercise 2.1 on p. 86 Exercise 2.2 on p. 88 Further Exercises 2 and 4 on p. 119
Group Theory
Simon Rubinstein-Salzedo
November 19, 2005
0.1
Introduction
These notes are based on a graduate course in group theory I took from Dr. Ken
Goodearl in the fall of 2004. The primary textbook for that class was Algebra by Larry
Grove. Some mate
Math 220C: Modern Algebra
Simon RubinsteinSalzedo
June 16, 2004
0.1
Introduction
Professor: Adebisi Agboola
Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30.
1
Chapter 1
Preliminaries
Question How can we tell whether a polyno
Math 220B: Modern Algebra
Simon RubinsteinSalzedo
March 11, 2004
0.1
Introduction
Professor: Adebisi Agboola
Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30.
1
Chapter 1
Rings
Denition 1.1 A ring (R, +, ) is an abelian group
Problem Set #7 Solutions Clifford Algebras For the first 4 problems, let R be a ring with 1 and let Mn (R) be the ring of n n matrices with entries in R. The set Rn of column vectors of length n is then an (Mn (R), R)-bimodule, and its transpose t R
Math 220C Spring 2008
Name Midterm Exam Round Two May 5, 2008
Instructions: (1) In round two of the midterm, you may supplement the answers you gave during the in-class exam in any way you choose. This is an open-book exam, but you may not consult
Problem Set #2 Solutions Chapter III. 2.2. 1. If F E K and E is stable, then GE G = G(K : F ). Proof. Let GE, G, and a E. Since E is stable, (a) E, and so (a) = (a). Thus -1 (a) = -1 (a) = a, so -1 GE. 2. If H G, then FH is stable. Proof. L
May 12, 2008 Math 220C Problem Set #5 Due: Monday, May 19 Grove, Chapter VI: Exercise Exercise Exercise Exercise Exercise 8.1 8.2 8.3 8.6 8.7 on on on on on p. 261 pp. 264265 p. 265 p. 270 p. 271
Math 220C Clifford Algebras
June 4, 2008
Let K be a field of characteristic different from 2, let V be a vector space over K, and let Q be a quadratic form on V , that is, a map Q : V K such that Q(v) = 2 Q(v) for all K, v V . There is an assoc
Math 220C Clifford Algebras, Part II
June 6, 2008
If Q is a nondegenerate quadratic form on a finite-dimensional C-vector space V , then in an appropriate basis B(ei , ej ) = ij . If dimC V = n, then the corresponding Clifford algebra will be denot
Problem Set #6 Solutions Chapter 5. 1.2. Let G = {1, x, x2 } be a cyclic group of order 3 and let R = Z2 . Write out addition and multiplication tables for the group algebra RG. We give a partial addition table. Note RG = {0, 1, x, x2 , 1 + x, 1 + x2
April 7, 2008 Math 220C Problem Set #2 Due: Wednesday, April 16, in Charlie Beil's mailbox (South Hall 6623) Grove, Chapter III: Exercise 2.2 on p. 88 (postponed from last week) Exercise 3.1 on p. 90 Exercise 3.2 on p. 91 Exercise 3.3 on p. 95
June 4, 2008 Math 220C Problem Set #7 Due: Friday, June 6 (but extensions will be available.) Let R be a ring with 1, and let Mn (R) be the ring of n n matrices with entries in R. The set Rn of column vectors of length n is then an (Mn (R), R)-bimod
Math 220C Midterm Solutions
Spring 2008
1. Let L be a subfield of complex numbers C such that [L : Q] = 2. Let Q the such that Q and Q( ) = L. Let M be the smallest subfield of C containing both Q( ) and L. Show that [M : Q] = 4. How many int
Math 220C Some character tables (revised)
May 19, 2008
In class, we have computed several character tables. (1) Character table for S3 . 1 (12) (123) |K| 1 3 2 trivial 1 1 1 1 det 1 -1 2 0 -1 V (2) Character table for BD4 . 1 -1 x y xy |K| 1 1 2 2
Math 220C Solving the cubic equation We want to find the roots of the general cubic polynomial f (x) = x3 + ax2 + bx + c F [x],
April 25, 2008
where F is a field containing a nontrivial cube root of unity . (That is, 3 = 1 but = 1.) We also assu
Problem Set #4 Solutions Chapter III. 5.1.i. Write out the elementary symmetric polynomials for n = 3, 4, 5. Using the form given on p. 103, k =
1i1 <i2 <ik n
xi1 xi2 xik ,
we do the case n = 5: 1 2 3 4 5 x1 + x2 + x3 + x4 + x5 x1 x2 + x1 x3 + x
Math 220A: Modern Algebra
Simon RubinsteinSalzedo
January 31, 2004
0.1
Introduction
Professor: Adebisi Agboola.
Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30.
Textbooks: Abstract Algebra by Dummit and Foote and Algebra by