Math 7752, Final Exam. Spring 2010. Due Tuesday, May 4th.
Directions: There are seven problems, each of which is worth 10 points.
The best six scores will be counted with 100% weight and the lowest score
will be counted with 50% weight, so the maximal tot
20. Galois correspondence
20.1. Further characterization of Galois extensions.
Proposition 20.1. Let K/F be a eld extension.
(a) K/F is Galois if and only if K is a splitting eld (over F ) for some
family of separable polynomials F [x].
(b) Assume that K/
12. Jordan canonical form of matrices and linear
transformations.
Let F be a eld, V a f.d. vector space over F and T gl(V ) an F -linear
transformation from V to V . As before let VT denote V considered as F [x]module where x acts as T .
Recall that the e
19. Galois groups and Galois extensions
Denition. Let K/F be a eld extension. Let Aut(K/F ) denote the set of
all F -automorphisms of K , that is,
Aut(K/F ) = cfw_ Aut(K ) : |F = idF .
Then Aut(K/F ) is clearly a group, called the automorphism group of K/
18. Separable extensions (continued)
We start with an example of a non-separable extension.
Example 18.1. Let Fp be a ntie eld of order p, let K = Fp ( ) be the eld
of rational functions over Fp in a formal variable and F = Fp ( p ). Then
it is easy to se
17. Normal and separable extensions
In this lecture we shall use a slightly generalized version of the Main Extension Lemma. The proof remains the same.
Lemma 17.0 (Generalized Main Extension Lemma). Suppose we are given
an algebraic extenstion K/M , an a
16. Algebraic closures and splitting fields
16.1. Uniqueness of algebraic closures.
Notation: Let : K L be a eld embedding. Then naturally extends
to a ring homomorphism : K [x] L[x] given by
(an xn + . . . + a0 ) = (an )xn + . . . + (a0 ).
Lemma 16.1 (S
15. Algebraic closures
We begin with one more result on algebraic extensions.
Lemma 15.1. Suppose that K/F and L/K are algebraic extensions. Then
the extension L/F is also algebraic.
Proof. In the case of nitely generated extensions this is immediate by
L
24. Solvability of equations by radicals and solvability of
Galois groups
The goal of this lecture is to prove the following theorem:
Theorem 24.1. Let F be a eld of characteristic zero, f (x) F [x] and K
a splitting eld for f (x) over F . Then the equati
22. Finite fields II
22.1. Main structure theorems. Recall that F is a eld of characteristic
p > 0, then the subeld of F generated by 1 (also called the prime subeld of F )
is isomorphic to Fp , so F is an extension of Fp .
The following results have been
Midterm #1, Spring 2010. Due Monday, March 1st.
Directions: There are ve problems, each of which is worth 10 points. All
problems will be counted. Provide complete arguments (do not skip steps).
State clearly any result you are referring to. Partial credi
Midterm #2, Spring 2010. Due Thursday, April 8th.
Directions: There are six problems, each of which is worth 10 points. The
best ve of six problems will be counted, but you are encouraged to do all
six. Provide complete arguments (do not skip steps). Stat
Midterm #2, Spring 2012. Due Thursday, April 12th.
Directions: There are ve problems, each of which is worth 10 points. The
best 4 out of 5 problems will be counted. Provide complete arguments (do
not skip steps). State clearly any result you are referrin
Midterm #1, Spring 2012. Due Thursday, March 1st.
Directions: There are ve problems, each of which is worth 10 points. The
best 4 out of 5 problems will be counted. Provide complete arguments (do
not skip steps). State clearly any result you are referring
23. Cyclic extensions
Problem. Given a eld F , describe all nite Galois extensions K/F with
Gal(K/F ) cyclic.
In this lecture we shall obtain a partial solution to this problem.
23.1. Linear independence of characters.
Denition. Let G be a group and L a e
26. Direct and inverse limits
26.1. Direct limits.
Denition. A poset A is called a directed set if for any , A there
exists A s.t. and .
Denition. Let C be a category. A direct system in C consists of a directed
set A, a collection of objects cfw_X A of C
25. Some category theory
25.1. Categories. A category C consists of the following data:
objects Ob(C )
for any X, Y Ob(C ) a set M or(X, Y ) = M orC (X, Y ) called morphisms from X to Y
for any triple X, Y, Z Ob(C ) a map
M or(X, Y ) M or(Y, Z ) M or(X
21. Galois correspondence (continued)
21.1. More on Galois correspondence. Our rst result provides yet another characterization of nite Galois extensions.
Proposition 21.1. Let K/F be a nite extension. Then K/F is Galois
K Aut(K/F ) = F .
Remark: The for
14. Field theory
Recall that a eld is a commutative ring with 1 in which all elements are
invertible.
14.1. Field extensions.
Denition. A eld extension is a pair of elds (K, F ) where K contains F .
The standard notation for a eld extension is K/F .
Denit
10. Canonical forms of linear transformations and similarity
classes of matrices.
Let F be a eld, V a vector space over F , n = dim V and gl(V ) =
HomF (V, V ). Note that gl(V ) M atn (F ) as F -algebras because of the
=
way matrix multiplication is dened
8. Modules over PID, part II. Smith Normal Form.
8.1. Proof of the Smith Normal Form theorem.
Theorem (Smith Normal Form (SNF). Let R be a PID, k, n N and
A M atkn (R). Then A can be written as A = CDB where B GLn (R),
a1
0
.
.
.
0
C GLk (R) and D M atkn
Homework Assignment # 8.
Plan for the next week: Galois correspondence (online lectures 20 and
21, Section 14.2 in DF), maybe start nite elds (online lecture 22, Section
14.3 in DF).
Problems, to be submitted by Thu, March 29th.
Problem 1: Prove the inter
Homework Assignment # 9.
Plan for the next week: Finite elds (online lecture 22, the end of 13.5
and 14.3 in DF), cyclic Galois extensions (online lecture 23, Section 6.6 in
Lang) and Solvability of equation by radicals (briey, online lecture 24, 14.7
in
Homework Assignment # 6.
Plan for next week: Normal and separable extensions (online lectures
17,18, 13.5 and parts of 13.4 in DF)
Problems, to be submitted by Thursday, March 15th.
Problem 1: V be a vector space over an algebraically closed eld F , let
n
Homework Assignment # 7.
Plan for next week: start Galois theory (online lecture 19, DF 14.1 and
parts of 14.2)
Problems, to be submitted by Thursday, March 22nd.
Problem 1: Let F be a eld and a subset of F [x]. Use the existence and
uniqueness of algebra
Homework Assignment # 4.
Plan for next week: Rational Canonical Form (12.2, online lectures 10-11).
Problems, to be submitted by Thu, February 16th.
Problem 1. (a) Let R be a commutative ring, let M be an R-module and
N its submodule. Prove that M is Noet
Homework Assignment # 5.
Plan for next week: Jordan canonical form (12.3 and online lecture 12+)
and eld extensions (13.1, 13.2 and online lecture 14).
Problems, to be submitted by Thu, February 23rd.
Problem 1: Let R be a commutative ring (with 1).
(a) L
Homework Assignment # 2.
Plan for next week: Tensor, symmetric an exterior algebras (11.5, online
lecture 6), modules over PID (12.1, online lecture 7).
Problems, to be submitted by Thu, February 2nd.
Problem 0. Read the section on graded algebras from on
Homework Assignment # 3.
Plan for next week: Modules over PID continued (12.1, online lectures
7-9).
Problems, to be submitted by Thu, February 9th.
Problem 1. The main goal of this problem is to classify 2-dimensional
R-algebras (R=reals), that is, R-alg