MA 553
GOINS
ASSIGNMENT #2
Due Friday, January 25 at the start of lecture.
Problem 1. Let (G, ) be a group. Let H and K be subgroups of G.
a. Show that the intersection H K is also a subgroup of G.
b.
MA 553 LECTURE NOTES: MONDAY, APRIL 7
Basic Theory of Field Extensions
Denitions. Let (R, +, ) be an integral domain. Recall that the characteristic of R, denoted by ch(R),
is that nonnegative integer
MA 553 LECTURE NOTES: FRIDAY, APRIL 4
Polynomial Rings over Integral Domains (contd)
Examples. Let (R, +, ) be a Unique Factorization Domain, and consider a monic polynomial f (x) R[x].
This has a roo
MA 553 LECTURE NOTES: WEDNESDAY, APRIL 2
1. Polynomial Rings over Fields (contd)
Polynomial Rings as Unique Factorization Domains. Recall that an integral domain (S, +, ) is a
unique factorization dom
MA 553 LECTURE NOTES: MONDAY, MARCH 31
Polynomial Rings
Denitions. Let (R, +, ) be a ring with 1 = 0. If x is an indeterminate, recall that a nite formal sum
p(x) = ad xd + + a1 x + a0 for ai R is cal
MA 553 LECTURE NOTES: FRIDAY, MARCH 28
Principal Ideal Domains (continued)
Proposition 1. Let (R, +, ) be an integral domain.
(1) If R is a Euclidean Domain, then R is a PID.
(2) If R is a PID, then R
MA 553 LECTURE NOTES: WEDNESDAY, MARCH 26
Euclidean Domains (contd)
In the last lecture we stated the following:
Proposition 1 (Euclidean Algorithm). Let (R, +, ) be a Euclidean Domain. Then the follo
MA 553 LECTURE NOTES: MONDAY, MARCH 24
Chinese Remainder Theorem (contd)
Example. Let R = Z. Choose positive integers m1 , m2 , . . . , mk , and let Ai = mi Z be proper ideals in R.
These ideals are p
MA 553 LECTURE NOTES: FRIDAY, MARCH 21
Chinese Remainder Theorem (continued)
Proposition 1 (Formosa Theorem). Let m1 , m2 , . . . , mk be positive integers which are relatively
prime in pairs. Let a1
MA 553 LECTURE NOTES: WEDNESDAY, MARCH 19
Rings of Fractions
Denitions. Let (R, +, ) be a commutative ring with 1 = 0. We say that a subset D R is a multiplicative
subset of R if
(i) 1 D; and
(ii) x y
MA 553 LECTURE NOTES: MONDAY, MARCH 17
Ring Isomorphism Theorems (contd)
Second Isomorphism Theorem. The following proposition shows how to place subrings and ideals in a
diamond lattice.
Proposition
MA 553 LECTURE NOTES: FRIDAY, MARCH 7
Group Rings (contd)
Proposition 1. Let (R, +, ) be a ring with 1 = 0, and (G, ) be a group. Dene binary operations
+, : R[G] R[G] R[G] by
=
+ =
ak gk
k
=
(ak + bk
MA 553 LECTURE NOTES: WEDNESDAY, MARCH 5
Examples of Rings
Subrings. Let (R, +, ) be a ring with 1 = 0. We say that S is a subring of R if the following axioms hold:
SR1 (Containment): S R.
SR2 (Addit
MA 553 LECTURE NOTES: MONDAY, MARCH 3
Introduction to Rings
Denitions. A ring is a triple (R, +, ) consisting of a set R and two binary operations + and on R
satisfying the following ve axioms:
(Addit
MA 553 LECTURE NOTES: WEDNESDAY, FEBRUARY 27
Recognizing Direct Products (contd)
Example. Let (G, ) be a group of order |G| = 10 = 2 5. Any Sylow 5-subgroup A has index |G : A| = 2,
so it must be norm
MA 553 LECTURE NOTES: MONDAY, FEBRUARY 25
Fundamental Theorem of Finitely Generated Abelian Groups (contd)
Elementary Divisors from Invariant Factors. We explain how to recover the Invariant Factors f
MA 553 LECTURE NOTES: FRIDAY, FEBRUARY 22
Fundamental Theorem of Finitely Generated Abelian Groups
Denitions. Let (G, ) be a group. Given a subset A G, recall that the subgroup of G generated by A is
MA 553 LECTURE NOTES: WEDNESDAY, FEBRUARY 20
The Simplicity of An
We prove that the alternating group of degree n is a simple group for suciently large n.
Proposition 1. An is simple for all n 5.
Proo
MA 553 LECTURE NOTES: WEDNESDAY, APRIL 9
Basic Theory of Field Extensions (contd)
Existence and Uniqueness of Simple Extensions. We explain how to generate eld extensions.
Proposition 1. Let (R, +, )
MA 553 LECTURE NOTES: FRIDAY, APRIL 11
Algebraic Extensions (contd)
Minimal Polynomials. Let be an element that is algebraic over a eld F . We discuss the nite extension
F () generated by .
Propositio
MA 553 HOMEWORK ASSIGNMENT #2 SOLUTIONS
Problem 1. Let (G, ) be a group. Let H and K be subgroups of G.
a. Show that the intersection H K is also a subgroup of G.
b. Show that the union H K is a subgr
MA 553
GOINS
ASSIGNMENT #1
Due Friday, January 18 at the start of lecture.
Problem 1. Let G be a set satisfying the following four axioms:
A1: There exists a binary operation : G G G i.e., a b G for a
MA 553 FINAL EXAM SOLUTIONS
Problem 1. Let (R, +, ) be an integral domain. The height of a prime p R is the largest integer d such
that there is a chain of distinct primes (0) = p0 p1 pd = p. Show tha
MA 553
GOINS
SAMPLE FINAL EXAM
This exam is to be done in two hours in one continuous sitting. Collaboration is not allowed.
You may not use your own personal notes; the lecture notes posted on the co
MA 553 SAMPLE FINAL EXAM SOLUTIONS
Problem 1. Let (R, +, ) be a commutative ring with 1 = 0. Show that the following are equivalent for a
proper ideal M R:
i. M is a maximal ideal.
ii. For each a R M
MA 553
GOINS
FINAL EXAM
This exam is to be done in two hours in one continuous sitting. Collaboration is not allowed.
You may not use your own personal notes; the lecture notes posted on the course we
MA 553 MIDTERM EXAM SOLUTIONS
Problem 1. Let (G, ) be a non-abelian simple group.
a. Show that its center Z (G) is trivial. [10 points]
b. Show that G is perfect i.e., it is generated by its commutato
MA 553
GOINS
SAMPLE MIDTERM EXAM
This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not allowed.
You may not use your own personal notes; the lecture notes posted on the
MA 553 MIDTERM SAMPLE EXAM SOLUTIONS
Problem 1. Let (G, ) be a group, and H G be a subgroup.
a. Show that NG (H )/CG (H ) is isomorphic to a subgroup of Aut(H ). [15 points]
b. Show that CG (H ) = NG
MA 553
GOINS
MIDTERM EXAM
This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not allowed.
You may not use your own personal notes; the lecture notes posted on the course