Math 100C: Abstract Algebra III
Spring 2017

Instructor's Office: 7250 APM
Instructor's Office Hours: 10:00-11:00am MWF
Lecture Time: 1:00-1:50pm MWF
Lecture Room: B412 APM
TA: Daniel Copeland
TA's Email: drcopela (at) ucsd.edu
TA's Office: 5801 APM
TA's Office Hours: 11:00am - 1:00pm W
Discussion Time: 6:00-6:50pm Tu
Discussion Room: 7421 APM
Final Exam Time: Thursday, 6/15/2017, 11:30am - 2:29pm
Final Exam Room: B412 APM

Lectures:

4/3: Administrivia. Section 10.1. Matrix representations of groups. Examples: Trivial representation, defining representation of dihedral groups, defining representation of S_n (review of presentations, permutations). Faithful representations.
4/5: Section 10.1. Characters and class functions. G-modules (vector spaces V with actions of groups). Equivalence of finite-dimensional G-modules and matrix representations. G-module homomorphisms. G-module homomorphisms in terms of matrices.
4/7: Section 10.2. G-submodules W of G-modules V. Examples: the circle group S^1 and the symmetric group S_n. Irreducible representations. Example (steering wheel circle action on R^2) and non-example. Action of a group G on a set S. Examples: action of the symmetric group S_n on k-element subsets of {1, ... , n}, action of the dihedral group D_n on diagonals in a regular n-gon, action of a group G on itself by left multiplication, action of a group G on itself by conjugation. Orbits and stabilizers. Statement of the Orbit-Stabilizer Theorem. Orbits and stabilizers for D_n action on diagonals.
4/10: Given a set S and a field k, the construction of k[S]. Permutation representations of groups. The group algebra k[G]. The regular representation. Overview of the Jordan Canonical Form.
4/12: Chapter 10. Direct sums (internal and exernal) of k-vector spaces. Direct sums (internal and external) of G-modules. Matrix version of direct sum: diagonal block matrices. Examples: circle group and S_3. Maschke's Theorem: If G is a finite group and k is a field with |G| not equal to 0 in k and V is a finite-dimensional G-module, then V is a direct sum of irreducible G-modules. Failure of Maschke's Theorem when G is infinite or when |G| = 0 in k. Proof of Maschke's Theorem by averaging projections.
4/14: Section 10.7. Review of G-module homomorphisms. "Baby" Schur's Lemma. The vector space Hom_G(V,W) and the algebra End_G(V). "Real" Schur's Lemma over an algebraically closed field. "Real" Schur is not true over R. Niceness Assumptions: All groups G are finite, all modules V are finite-dimensional, over C. Section 10.3. Inner products on complex vector spaces. Statement: "If V is a G-module, V admits a G-invariant inner product." Supplemental material on G-module homomorphisms.
4/17: Section 10.3. Proof that G-modules admit G-invariant inner products. Conjugate transpose of matrices. Unitary matrices. Unitary group. With respect to an appropirate basis, any representation maps into unitary matrices. If chi is a character and g is in G, then chi(g^(-1)) is the complex conjugate of chi(g). Section 10.4. Notation for characters. Class functions. Class function inner product.
4/19: Section 10.4. Statement of character orthogonality (Main Theorem of Rep'n Theory of finite groups over C). Corollaries: The number of non-isomorphic irreducible G-modules is the number of conjugacy classes of G. Two representations are isomorphic iff they have the same character. Decompositions into irreducibles is unique. A representation V with character chi is irreducible iff < chi, chi > = 1. The multiplicity of an irreducible representation V_i in any representation V is < chi_i, chi >, where chi_i is the character of V_i and chi is the character of V. Character tables. The character table of S_3.
4/21: Section 10.4/10.5. Decomposing the action of S_3 on set partitions of {1,2,3} into irreducibles. The character table of C_4. Decomposing the regular representation C[G] into irreducibles. The Magic Formula: the sum of the dimensions of the non-isomorphic G-irreps squared equals the order of G.
4/24: Chapter 10. The character table of D_4. Character orthogonality of the second kind. The character table of S_4. Contruction of the 2-dimensional S_4 irrep. with baseballs.
4/26: Section 10.8. Proof of Character Orthogonality of the First Kind.
4/28: Midterm 1.
5/1: Section 15.1. Review of fields. Examples. Field extensions K/F. Section 15.2. Algebraic and transcendental elements in field extensions K/F. Evaluation homomorphism. The irreducible polynomial of an element a in K over F. Examples of irreducible polynomials.
5/3: Section 15.2. If K/F is an extension and a_1, ... , a_n are in K, we have F[a_1, ... , a_n] and F(a_1, ... , a_n). If the a_i are algebraic over F, then F[a_1, ... , a_n] = F(a_1, ... , a_n). F-vector space basis for F(a). F-isomorphisms of two extensions K/F and K'/F. Building F-isomorphisms between extensions of the form F(a). Examples (including complex conjugation). Section 15.3. The degree [K:F] of an extension K/F. If a is algebraic over F, then [F(a):F] is the degree of the irreducible polynomial of a over F. Algebraic and finite extensions.
5/5: Section 15.3. If K/L and L/F are extensions, then [K:F] = [K:L]*[L:F]. If K/F is an extension and a is in K, then a is algebraic over F iff [F(a):F] is finite. Every finite extension is algebraic. Not every algebraic extension is finite. If a_1, ... , a_n in K are algebraic over F, then F(a_1, ... , a_n)/F is algebraic. Corollary: A sum or product of algebraic elements over F remains algebraic over F.
5/8: Section 15.6. If f is an irreducible monic polynomial in F[x], then F[x]/(f) is a field containing F over which f has a root. If f(x) is any polynomial in F[x], there is a field extension K of F over which f(x) splits completely into linear factors. Separable polynomials. The derivative f' of a polynomial f over a field F[x]. A nonconstant polynomial f in F[x] is separable iff gcd(f,f') = 1. If f in F[x] is irreducible, then f is separable unless f' = 0. Every irreducible polynomial over a field of characteristic zero is separable. Example of an irreducible polynomial over F_p(t) which is not separable. Section 15.7. Intro to finite fields. Finite fields have prime power orders.
5/10: Construction of F_4 and F_8. If q is a power of a prime p, there exists a field K of order q (which has characteristic p and contains a copy of F_p). Any two fields of order q are isomorphic.
5/12: If p^r is a prime power and K is a field of order p^r, then K contains a subfield of order p^s iff s|r, in which case this subfield is unique. The lattice of p-power fields. Section 15.8. Primitive elements of field extensions. Example: Q(sqrt(3), sqrt(5))/Q. Primitive Element Theorem: If K/F is a finite extension of fields in characteristic zero, there exists a primitive element in K for this extension. Proof of the Primitive Element Theorem.
5/15: Section 16.1. The action of S_n on F[x_1, ... , x_n] by variable permutation. Symmetric polynomials. Elementary symmetric polynomials s_1, s_2, ... , s_n. Newton's Theorem: Every symmetric polynomial in F[x_1, ... , x_n] is a polynomial, with coefficients in F, in the elementary symmetric polynomials s_1, s_2, ... , s_n. Corollary: Let f(x) in F[x] be a polynomial and let K/F be a field extension over which f(x) splits completely with roots a_1, a_2, ..., a_n. Then any symmetric function g(a_1, ... , a_n) in the roots of f(x) lies in the base field F.
5/17: Section 16.3. Splitting fields of polynomials. Example: The splitting field of x^3 - 2 over Q. Splitting fields are finite extensions. A field extension contains at most one splitting field of a given polynomial. The Splitting Theorem: If K/F is the splitting field for f(x) over F, and g(x) is an irreducible polynomial in F[x] with a single root in K, then g(x) splits completely in K[x]. Proof of the Splitting Theorem.
5/19: Section 16.4. F-isomorphisms of extensions K/F and K'/F. The Galois group G(K/F) of an extension. Galois extensions. C/R and Q(sqrt(2))/Q are Galois. Q(alpha)/Q is not Galois, where alpha is a real cube root of 2. If f(x) is a polynomial in F[x], then any two splitting fields for f(x) over F are F-isomorphic.
5/22: Section 16.5. Fixed fields. If H is a finite subgroup of Aut(K) and F = K^H, then for beta in K the degree of beta over F equals the size of the H-orbit of beta. K/F is finite of degree |H|. Section 16.6. If K/F is any finite extension, then |G(K/F)| divides [K:F]. If H is any finite subgroup of Aut(K), then K/K^H is a finite Galois extension with Galois group H.
5/24: Section 16.6. A fintie extension K/F with Galois group G is Galois iff K^G = F iff K is the splitting field for some polynomial in F[x]. If K/L/F is a tower of extensions with K/F finite and Galois, then K/L is Galois (but L/F is not necessarily Galois). Section 16.7. The Fundamental Theorem of Galois Theory: Galois correspondence. Examples: splitting fields over Q of x^3 - 2 and (x^2 - 3)(x^2 - 5).
5/26: Midterm 2.
5/31: Section 16.7. If K/F is a Galois extension and L is an intermediate field, then L/F is Galois iff G(K/L) is a normal subgroup of G(K/F). Section 16.10. Cyclotomic extensions. All cyclotomic extensions are Galois. If p is prime and zeta = exp(2*pi*i/p), then Q(zeta)/Q is Galois with Galois group cyclic of order p-1. Moreover, F(zeta)/F is Galois with cyclic Galois group for any subfield F of C. Statement of results related to cyclotomic polynomials and unit group of Z/nZ.
6/2: Section 16.11. Kuemmer extensions. If F is a subfield of C containing all the nth roots of unity, and a^n is in F, then F(a)/F is a Galois extension with cyclic Galois group. Solvable groups. Examples: finite abelian groups, D_n, S_4. Non-examples: A_n, S_n for n > 4. Fact: If F is a subfield of C, the splitting field K over F of any polynomial of the form (x^(p_1) - a_1) ... (x^(p_n) - a_n) with p_i prime and a_i in F has G(K/F) solvable. Section 16.12. Cardano's formula for the cubic. Extensions of fields by radicals. Solvable polynomials. Statement of Main Theorem: Let f(x) in Q[x] be a polynomial which is solvable by radicals. The Galois group of f(x) is a solvable group. Proof that f(x) = x^5 - 16*x + 2 has Galois group S_5, so that f(x) is not solvable by radicals.
6/5: Quotients of solvable groups are solvable. If G is a finite group, N is a normal subgroup of G, and N and G/N are solvable, then G is solvable. Section 16.12. Proof of the Main Theorem: Let f(x) in Q[x] be a polynomial which is solvable by radicals. The Galois group of f(x) is a solvable group. Section 9.1. Real and complex linear groups. Examples.
6/7: Section 9.4. One-parameter subgroups of linear groups G. Examples on the torus. The exponential e^X of a matrix X. One-parameter subgroups of GL_n all have the form f(t) = e^(tX). Section 9.5. The Lie algebra of a linear group. The Lie algebras of GL_n, SL_n, SO_n.

Homework Assignments:

Homework 1, due 4/12/2017. LaTeX file. Solutions.
Homework 2, due 4/19/2017. LaTeX file. Solutions.
Homework 3, due 4/26/2017. LaTeX file. Solutions.
Homework 4, due 5/10/2017. LaTeX file. Solutions.
Homework 5, due 5/17/2017. LaTeX file. Solutions.
Homework 6, due 5/24/2017. LaTeX file. Solutions.
Homework 7, due 6/7/2017. LaTeX file. Solutions.

Midterms:

Practice Midterm 1.

Midterm 1.
Score Distribution: 100, 100, 99, 96, 95, 93, 90, 89, 87, 82, 75, 74, 71, 68, 59, 58, 58, 51, 33, 20.

Practice Midterm 2.

Midterm 2.
Solutions and Comments. Score Distribution: 98, 97, 90, 90, 89, 84, 84, 84, 82, 81, 75, 72, 69, 65, 64, 63, 60, 57, 43, 39, 15.

Practice Final. Solutions (the last two pages are switched -- sorry!).