Spring 2017

**Instructor:** Brendon Rhoades

**Instructor's Email:** bprhoades (at) math.ucsd.edu

**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

** Syllabus:**
Please read carefully.

** 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.

** Some Lecture Notes:**

Lecture 3

Lecture 4

Lecture 5

Lecture 6

Lecture 7

Lecture 8

Lecture 9

Lecture 10

Lecture 11

Lecture 12

Lecture 13

Lecture 14

Lecture 15

Lecture 16

Lecture 17

Lecture 18

Lecture 19

Lecture 20

Lectures 21/22

Lecture 23

Lecture 24

Lectures 25/26

** 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!).