School of Mathematics School of Mathematics
Course 111 - Algebra 2003-2004 (JF Mathematics, Theoretical Physics & Two-subject Moderatorship )
Lecturer: Dr. C. Ó Dúnlaing
Requirements/prerequisites: None

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: The natural number system and Peano's axioms
The integers , divisibility, and congruence modulo n
Remainder modulo n and integer division
Semigroups, monoids, and groups
Groups
Additive subgroups of Z
The symmetric group Sn
Generators for Sn
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 Zn*
First isomorphism theorem for groups
Prime factorisation theorem
A Sylow theorem
Rings
Zero divisors, integral domains, and fields.
Ring homomorphisms
Characteristic of a ring (omitted)
Polynomials
Division algorithm for polynomials over a field
Factorising polynomials
Gauss's Lemma and Eisenstein's Criterion
Ring homomorphisms and ideals
Principal ideal domains
Dimension of extension fields
Ruler-and-compass constructions
Cubic equations
The Galois group of an extension field
Normal field extensions
Stable intermediate fields
Splitting fields
Radical field extensions and solvability
A polynomial equation not solvable by radicals
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 TEX by TTH, version 2.70.
On 14 Jun 2004, 15:47.