TENSOR PRODUCTS
KEITH CONRAD
1. Introduction
Let R be a commutative ring and M and N be R-modules. (We always work with rings
having a multiplicative identity and modules are assumed to be unital: 1 m
TENSOR PRODUCTS II
KEITH CONRAD
1. Introduction
Continuing our study of tensor products, we will see how to combine two linear maps
M M and N N into a linear map M R N M R N . This leads to at modules
TRACE AND NORM
KEITH CONRAD
1. Introduction
Let L/K be a nite extension of elds, with n = [L : K ]. We will associate to this
extension two important functions L K , called the trace and the norm.
For
ZORNS LEMMA AND SOME APPLICATIONS
KEITH CONRAD
1. Introduction
Zorns lemma is a result in set theory which appears in proofs of some non-constructive
existence theorems throughout mathematics. We will
Math 5211 - Abstract Algebra II
Due in MSB 318
Problem Set 1
2/1/13 at 4 PM
Much research for new proofs of theorems already correctly established is undertaken
simply because the existing proofs have
Math 5211 - Abstract Algebra II
Due in MSB 318
Problem Set 2
2/15/13 at 4 PM
Mathematical education is still suering from the enthusiasms which the discovery of this
isomorphism [Homk (V, V ) = Mn (k
Math 5211 - Abstract Algebra II
Due in MSB 318
Problem Set 3
3/1/13 at 4 PM
One cannot escape the feeling that these mathematical formulas have an independent
existence and intelligence of their own,
Math 5211 - Abstract Algebra II
Due in MSB 318
Problem Set 4
4/12/13 at 4 PM
Often the signicance of a mathematical theorem becomes clear only when looked at from
above that is to say, from the standp
Math 5211 - Abstract Algebra II
Due in MSB 318
Problem Set 5
4/29/13 at 4 PM
I consider that I understand an equation when I can predict the properties of its solutions
without actually solving it.
P.
CONSTRUCTING ALGEBRAIC CLOSURES
KEITH CONRAD
Let K be a eld. We want to construct an algebraic closure of K , i.e., an algebraic
extension of K which is algebraically closed. It will be built as the q
SYMMETRIC POLYNOMIALS
KEITH CONRAD
Let F be a eld. A polynomial f (T1 , . . . , Tn ) F [T1 , . . . , Tn ] is called symmetric if it is
unchanged by any permutation of its variables:
f (T1 , . . . , Tn
STABLY FREE MODULES
KEITH CONRAD
1. Introduction
Let R be a commutative ring. When an R-module has a particular module-theoretic
property after direct summing it with a nite free module, it is said to
CYCLOTOMIC EXTENSIONS
KEITH CONRAD
1. Introduction
For any eld K , a eld K (n ) where n is a root of unity (of order n) is called a cyclotomic
extension of K . The term cyclotomic means circle-dividin
DUAL MODULES
KEITH CONRAD
1. Introduction
Let R be a commutative ring. For two (left) R-modules M and N , the set HomR (M, N ) of
all R-linear maps from M to N is an R-module under natural addition an
EXTERIOR POWERS
KEITH CONRAD
1. Introduction
Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules
and all tensor products are taken over R, so we abbreviate R to . A bili
FINITE FIELDS
KEITH CONRAD
This handout discusses nite elds: how to construct them, properties of elements in a
nite eld, and relations between dierent nite elds (including their Galois groups). We
wr
THE GALOIS CORRESPONDENCE
KEITH CONRAD
1. Introduction
We call a nite extension of elds L/K Galois if L is the splitting eld over K of a
separable polynomial: some (monic) separable polynomial f (X )
ROOTS AND IRREDUCIBLES
KEITH CONRAD
1. Introduction
This handout discusses relationships between roots of irreducible polynomials and eld
extensions. Throughout, the letters K , L, and F are elds and
SEPARABILITY
KEITH CONRAD
1. Introduction
Let K be a eld. We are going to look at concepts related to K that fall under the label
separable.
Denition 1.1. A nonzero polynomial f (X ) K [X ] is called
SEPARABILITY II
KEITH CONRAD
1. Introduction
Separability of a nite eld extension L/K can be described in several dierent ways. The
original denition is that every element of L is separable over K (th
ISOMORPHISM OF SPLITTING FIELDS
KEITH CONRAD
Using tensor products, we will give a slick proof that any two splitting elds of a polynomial are (non-canonically) isomorphic over the base eld.
Theorem 1
COMPLEXIFICATION
KEITH CONRAD
1. Introduction
We want to describe a procedure for enlarging real vector spaces to complex vector spaces
in a natural way. For instance, the natural complex analogues of