Entry requirements

120 credit points including Algebraic Structures

Learning outcomes

At the end of the course the student should be able to

• define the concepts algebraic number, algebraic integer, Dedekind ring, fractional ideal, ideal class, ideal class group, regular prime, irregular prime, norm, trace, ramification index, residue class degree;
• give an account of the fundamental theorem of Dedekind rings;
• in elementary cases calculate the norm and trace of an algebraic number;
• explain how the ideal class number reflects the deviation of a ring of algebraic integers from being factorial;
• outline Kummer's lemma;
• outline Kummer's proof of Fermat's great theorem for regular prime exponents;
• outline the links between the ramification index and the residue class degrees of a prime ideal, with respect to an algebraic field extension.

Content

Integral elements in commutative ring extensions. Dedekind rings, their ideal theory and ideal class group. Quadratic number fields. Cyclotomic fields and cyclotomic integers. Regular and irregular primes. Kummer's lemma. Kummer's proof of Fermat's great theorem for regular prime exponents. Norm and trace of an algebraic number. Ramification index and residue class degree of a prime ideal with respect to an algebraic field extension.

Instruction

Lectures.

Assessment

At the end of the course written examination and, possibly, assignments and/or oral examination in accordance with instructions given at course start.