Algebra

(3-4 Credit Hours) · Undergraduate

"MATH 30100 Field Theory at UCD; This course gives an introduction to field theory, with an emphasis on the study of finite fields, which underlies many modern applications. Axioms, properties and constructions of arbitrary fields will be described. Fields will be viewed both as vector spaces and as quotients of polynomial rings. While the study of fundamental infinite fields will be given a rigourous treatment, students will also study properties of finite fields in a concrete way. Standard techniques used in the manipluation of the algebra of finite fields will be introduced. Applications of finite fields arise naturally in digital technology, such as coding theory, cryptography or sequences. Some of these applications will be described here. During the course of this module, students will:1. become familiar with the properties of a field;2. have a knowledge of constructions and properties of important classes of fields; 3. be furnished with an understanding of field theory so that further studies in algebra reliant on field theory can be undertaken, such as Galois theory, number theory and commutative algebra;4. have a concrete understanding of finite fields, including manipulation of additive and multiplicative representations of elements;5. be familiar with the Frobenius automorphism and trace map and their use as tools in the study of finite fields;6. be familiar with important applications of finite fields related to the mathematics of communications theory." When taught in Perth: The theme of this unit is the theory of groups, and their use in measuring symmetry with special emphasis on geometric examples. Apart from the mathematical interest of groups, they are of great and increasing importance in chemistry, geology and physics, and those aspects that have significance in such disciplines are strongly emphasised. The fundamental notion is that of a group action. Group actions are used to elucidate the structure of a group, culminating in the Sylow theorems, which connect finite groups and number theory. The applications of group actions extend to symmetries of the plane, three-space and higher dimensional spaces, rotation and spin, crystallographic groups and symmetries of regular figures in two and three dimensions. Other algebraic structures such as rings and fields may also be explored. When taught at Trinity College Dublin: Sets, their unions, intersections, differences, direct (or cartesian) products. Subsets. Maps between sets, injective, surjective and bijective maps. Images and preimages of subsets. Composition of maps. Identity map, the inverse of a map, right and left inverses. Binary operations on sets. Associativity, multiplicativity. Identity and inverse elements with respect to a binary operation. Groups, semigroups, monoids. Cayley table of a group. Direct products of groups. Subgroups. Intersections of subroups. Generators of a subgroup. Permutation group of a set (the group of all bijective self-maps). Symmetric group Sn. Parity (sign) of a permutation, even and odd permutations. Alternating subgroup An of Sn. Group of Isometries. Matrix groups GLn, SLn, On, SOn, Un, SUn. Integer division with remainder. Additive subgroups of Z. Greatest common divisor. Euclidean algorithm. Unique prime factorization. Binary relations, equivalence relations, partitions. Congruence relation and classes of integers modulo n. The set of congruence classes Zn modulo n as additive group and multiplicative monoid. Multiplicative group Zn*. Cosets of a subgroup in a group. Lagrange's Theorem. Group homomorphisms and isomorphisms. Kernel of homomorphism. Normal subgroup. Quotient group modulo normal subgroup. First isomorphism theorem. Chinese remainder theorem. Group actions on a set. Stabilizers and orbits of a group action. Sylow's Theorem. Classification of finite abelian groups.