**Duration:**

**Number of lectures per week:** 2 and 1 tutorial

**Assessment:** Continuous assessment through twenty homeworks and three
short examinations spread over the academic year.

**End-of-year Examination:** No annual examination. Those who fail through continuous
assessment must take a supplemental examination (in September)

**Description: **

Number theory:
The natural number system and Peano's axioms

The integers , divisibility, and congruence modulo n

Remainder modulo n and integer division

Groups:
Semigroups, monoids, and groups

Groups

Additive subgroups of **Z**

The symmetric group S_{n}

Generators for S_{n}

Parity and the alternating group

Binary relations, equivalence relations, and partitions

Cosets, Lagrange's Theorem, and Fermat's Theorem

Normal subgroups and quotient groups

Greatest common divisor

Multiplicative group **Z**_{n}^{*}

First isomorphism theorem for groups

Prime factorisation theorem

A Sylow theorem

Rings and fields:
Rings

Zero divisors, integral domains, and fields.

Ring homomorphisms

Characteristic of a ring

Polynomials

Division algorithm for polynomials over a field

Factorising polynomials

Gauss's Lemma and Eisenstein's Criterion

Ring homomorphisms and ideals

Principal ideal domains

Solving equations:
Dimension of extension fields

Ruler-and-compass constructions

Cubic equations

The Galois group of an extension field

Normal extensions, stable intermediate fields,
and splitting fields.

Certain standardised radical extensions have
solvable group; radical splitting fields have
solvable group.

A polynomial equation not solvable by radicals

A last result:
Finite multiplicative subgroups of a field

**Textbooks:**
John R. Durbin, *Modern algebra - an introduction*,
contains some but not all of the material.

Jun 14, 2004

File translated from T

On 14 Jun 2004, 15:46.