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. Show that the union H K is a subgroup if and only if e
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 m that that ker() = m Z is the kernel of the map : Z R
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 root R if and only if g (x) = x divides f (x). To be more
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 domain or a UFD if every nonzero s S has a unique factoriz
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 called a polynomial. We denote deg(p) = d as its degree if
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 is a UFD.
Proof. (2) Last time, we showed existence of
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 following are
equivalent for nonzero elements a, b R:
i. The
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 pairwise co-maximal if the mi are paiwise co-prime, beca
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 , a2 , . . . , ak be arbitrary integers. Then the syste
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 D for all x, y D.
Proposition 1. Let (R, +, ) be a com
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 1 (Diamond Isomorphism Theorem). Let (R, +, ) be a ring
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 ) gk
k
bk gk
=
k
ai bj gk
k
gi gj =gk
Then (R[G], +,
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 (Addition): (S, +) is a group with identity 0 and inverses a
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:
(Addition): (R, +) is a group with identity 0 and inverses a
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 normal in G. Let B be a Sylow 2-subgroup of G. We show that
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 from
the Elementary Divisor Decomposition. To this end,
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
the subgroup
H = a1 1 a2 2 an n
A=
k Z, ak A .
AH G
We
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.
Proof. This will be by induction on n. Consider the proposi
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, +, ) be an integral domain, and F be its eld of fractions. L
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 .
Proposition 1. Let (F, +, ) be a eld. Let be algebraic over F .
(
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 subgroup if and only if either H K or K H .
Solution:
(a.) W
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 all a, b G.
A2: Associativity holds i.e., (a b) c = a (b
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 that in a Principal Ideal Domain
every nonzero prime has h
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 course web site;
homework solutions posted on the course
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 there exists b R such that 1 a b M .
[10 points]
Soluti
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 web site;
homework solutions posted on the course web sit
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 commutators. [10 points]
Solution:
(a.) The center Z (G) G, so e
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 course web site;
homework solutions posted on the cour
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 (H ) if |H | = 2. [5 points]
Solution:
(a.) Let g NG (H
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 web site;
homework solutions posted on the course web