1
Topics in Group Theory
1.1
Groups
A binary operation on a set G associates to elements x and y of G a third
element x y of G. For example, addition and multiplication are binary
operations of the set of all integers.
Denition A group G consists of a set

5
5.1
Hilberts Nullstellensatz
Commutative Algebras of Finite Type
Denition Let K be a eld. A unital ring R is said to be a K -algebra
if K R, the multiplicative identity elements of K and R coincide, and
ab = ba for all a K and b R.
It follows from this

6
Introduction to Ane Schemes
6.1
Rings and Modules of Fractions
Let R be a unital commutative ring. A subset S of R is said to be a multiplicative subset if 1 S and ab S for all a S and b S .
Let M be a module over a unital commutative ring R, and let S

5
Simplicial Complexes
5.1
Geometrical Independence
Denition Points v0 , v1 , . . . , vq in some Euclidean space Rk are said to be
geometrically independent (or ane independent ) if the only solution of the
linear system
q
j =0 j vj = 0,
q
=0
j =0 j
is th

4
4.1
Covering Maps and Discontinuous Group
Actions
Covering Maps and Induced Homomorphisms of
the Fundamental Group
Proposition 4.1 Let p: X X be a covering map over a topological space
X , let : [0, 1] X and : [0, 1] X be paths in X , where (0) = (0) an

3
3.1
Covering Maps and the Monodromy Theorem
Covering Maps
Denition Let X and X be topological spaces and let p: X X be a
continuous map. An open subset U of X is said to be evenly covered by the
map p if and only if p1 (U ) is a disjoint union of open s

2
2.1
Homotopies and the Fundamental Group
Homotopies
Denition Let f : X Y and g : X Y be continuous maps between
topological spaces X and Y . The maps f and g are said to be homotopic if
there exists a continuous map H : X [0, 1] Y such that H (x, 0) = f

4
Commutative Algebra and Algebraic Geometry
4.1
Modules
Denition Let R be a unital commutative ring. A set M is said to be a
module over R (or R-module ) if
(i) given any x, y M and r R, there are well-dened elements x + y
and rx of M ,
(ii) M is an Abel

3
Introduction to Galois Theory
3.1
Field Extensions and the Tower Law
Let K be a eld. An extension L: K of K is an embedding of K in some
larger eld L.
Denition Let L: K and M : K be eld extensions. A K -homomorphism
: L M is a homomorphism of elds which

2
Rings and Polynomials
2.1
Rings, Integral Domains and Fields
Denition A ring consists of a set R on which are dened operations of
addition and multiplication satisfying the following axioms:
x+y = y +x for all elements x and y of R (i.e., addition is c

Course 311: Michaelmas Term 2005
Part I: Topics in Number Theory
D. R. Wilkins
Copyright c David R. Wilkins 19972005
Contents
1 Topics in Number Theory
1.1 Subgroups of the Integers . . . . . . . . . .
1.2 Greatest Common Divisors . . . . . . . . . .
1.3