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 = m for all
m M .) The direct sum M N is an addition o
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
and linear maps between base extensions. Then we will
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 each L, let m : L L be multiplication by : m (x) = x f
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 state Zorns lemma below and use it
in later sections t
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 no aesthetic appeal [.] An elegantly executed
proof is
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 )] has aroused. The result has been that geometry was
e
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, that they are wiser than we are, wiser even than
their
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 standpoint of a more advanced theory. But the meaning
is alwa
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. A. M. Dirac
Read: 13.6, 14.1, 14.2, 14.3, 14.5 or hand
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 quotient of a polynomial
ring in a very large number of
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 ) = f (T(1) , . . . , T(n) )
for all Sn .
Example 1. T
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 have the property
stably. For example, R-modules M and
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-dividing, and comes from the fact that the
nth roots of unity
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 and scaling operations
on linear maps. (If R were non-com
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 bilinear function
out of M1 M2 turns into a linear function
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
write Z/(p) and Fp interchangeably for the eld of size p.
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 ) K [X ] splits completely
over L and L is generated over
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 Fp = Z/(p) is the eld of
p elements. When f (X ) K [X ]
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 separable when it has distinct
roots in a splitting eld
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 (that is, has a separable
minimal poynomial in K [X ]). We
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. Let K be a eld and f (X ) K [X ] be nonconstant. Any
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 Rn , Mn (R), and R[X ]
are Cn , Mn (C) and C[X ].
Why