3704 (Algebraic Number Theory)
Old Code: MATHC392
Value: Half unit (= 7.5 ECTS credits)
Structure: 3 hour lectures per week
Assessment: 100% examination
Normal Pre-requisites: First two years algebra (1201/2, 2201, 7202)
Lecturer: Prof M Kim
Course Description and Objectives
This course aims to introduce students to some of the applications of algebra in number theory,
and starts the study of algebraic number theory, one of the two main branches of modern
number theory. The course builds on the algebra studied in earlier years. A recommended book
is: Stewart and Tall, Algebraic Number Theory and Fermat’s Last Theorem.
Field extensions, Primitive element theorem, Newton’s theorem on symmetric polynomials.
Algebraic numbers, algebraic integers, conjugates, discriminants, integral bases, norms and
traces. Cyclotomic and quadratic ﬁelds.
Non-unique factorization of algebraic integers, unique factorization into prime ideals, fractional
ideals, class group, norms of ideals.
Lattices and Minkowski’s Lemma, ﬁniteness of class number, calculation of class numbers.
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