Algebra -- Dr Neumann -- 24 lectures MT
Rings and arithmetic [8 lectures, Michaelmas Term]
Aims and Objectives
This half-course introduces the student to some classic
ring theory which is basic for other parts of
abstract algebra, for linear algebra and for
those parts of number theory that lead ultimately
to applications in cryptography.
The first-year algebra course contains a treatment of
the Euclidean Algorithm in its classical
forms for integers and for polynomial rings over
a field.
Here the idea is developed in abstracto.
The Gaussian integers, which have applications to
many questions of elementary number theory,
give an important and interesting (and entertaining)
illustration of the theory.
Synopsis
Commutative rings with unity, integral domains, fields;
examples including polynomial rings and subrings of R and C.
Ideals and quotient rings;
isomorphism theorems.
Examples: integers modulo a natural number;
the Chinese Remainder Theorem;
the quotient ring by a maximal ideal is a field.
Arithmetic in integral domains.
Units, associates, irreducible elements, primes.
Euclidean rings and their properties:
Division Algorithm, Euclidean Algorithm;
Z and F[x] as prototypes.
Properties of euclidean rings: their
ideals are principal;
their irreducible elements are prime;
factorisation is unique.
Examples for applications:
Gauss's Lemma and factorisation in Q[x];
factorisation of Gaussian integers.
Reading
PETER J CAMERON, Introduction to Algebra, OUP 1998,
ISBN 0-19-850194-3. Chapter 2.
Alternative and further reading:
JOSEPH J ROTMAN, A First Course in Abstract Algebra,
(Second edition), Prentice Hall, 2000, ISBN 0-13-011584-3.
Chapters 1, 3.
I N HERSTEIN, Topics in Algebra (Second edition) Wiley
1975, ISBN 0-471-02371-X. Chapter 3.
[Harder than some, but an excellent classic. Widely available
in Oxford libraries; still in print.]
P M COHN, Classic Algebra, Wiley 2000,
ISBN 0-471-87732-8. Various sections.
[This is the third edition of his book previously called
Algebra I.]
DAVID SHARPE, Rings and Factorization,
CUP 1987, ISBN 0-521-33718-6. [An excellent little book,
now sadly out of print; available in some libraries, though.]
There are very many other such books on abstract algebra in
Oxford libraries.
Further Linear Algebra [16 lectures in Michaelmas Term]
Aims and Objectives
The core of linear algebra comprises the theory of linear equations in
many variables, the theory of matrices and determinants, and the theory
of vector spaces and linear transformations. All these topics were
introduced in the Moderations course. Here they are developed further
to provide the tools for applications in geometry, modern mechanics and
theoretical physics, probability and statistics, functional analysis
and, of course, algebra and number theory.
Our aim is to provide a thorough treatment of some classical theory
that describes the behaviour of linear transformations on a
finite-dimensional vector space to itself, both in the purely algebraic
setting and in the situation where the vector space carries a metric
deriving from an inner product.
Synopsis.
Fields; Q, R, C, F_p as examples. Vector spaces over an arbitrary
field, subspaces, direct sums; projection maps and their characterisation
as idempotent operators.
Dual spaces of finite-dimensional spaces; annihilators; the
natural isomorphism between a finite-dimensional space and its second
dual; dual transformations and their matrix representation with respect
to dual bases.
Some theory of a single linear transformation on a
finite-dimensional space: characteristic polynomial, minimal polynomial,
Primary Decomposition Theorem, the Cayley-Hamilton theorem (economically);
diagonalisability; triangular form.
Real and complex inner product spaces: examples, including function spaces
[but excluding completeness and L^2]. Orthogonal complements, orthonormal
sets; the Gram-Schmidt process. Bessel's inequality; the Cauchy-Schwarz
inequality.
Some theory of a single linear transformation on a finite-dimensional
inner product space: the adjoint; eigenvalues and diagonalisability of
a self-adjoint linear transformation.
Reading
RICHARD KAYE and ROBERT WILSON, Linear Algebra, OUP 1998,
ISBN 0-19-850237-0. Chapters 2-13.
[Chapters 6, 7 are not entirely relevant to our syllabus, but are interesting.]
Alternative and further reading:
PAUL R HALMOS Finite-dimensional Vector Spaces,
(Reprint 1993 of the 1956 second edition),
Springer Verlag ISBN 3-540-90093-4. §§1-15, 18, 32-51, 54-56,
59-67, 73, 74, 79. [Now over 50 years old, this idiosyncratic
book is somewhat dated but it is a great classic, and well worth reading.]
P M COHN, Classic Algebra, Wiley 2000, ISBN 0-471-87732-8.
§§4.1-4.7, 7.1-7.3, 8.1-8.5. [Third edition of his earlier
*Algebra, I*. Does not cover the whole syllabus--but
does have relevant material from other parts of algebra.]
SEYMOUR LIPSCHUTZ AND MARC LIPSON, Schaum's Outline of
Linear Algebra (3rd edition, McGraw Hill 2000), ISBN
0-07-136200-2. [Many worked examples.]
C W CURTIS,
Linear Algebra--an Introductory Approach, Springer (4th
edition), reprinted 1994
D T FINKBEINER,
Elements of Linear Algebra, Freeman, 1972
[Out of print, but available in many libraries]