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, discriminant, ramification index, residue class grade;
• outline the ideal theory of Dedekind rings;
• in elementary cases calculate the norm and trace of an algebraic number;
• explain whether the ideal class number reflects the ideal class number's deviation from being factorial;
• outline Kummer's lemma;
• generally outline Kummer's proof of Fermat's great theorem for regular prime exponents;
• outline the theorem which links the ramification index and the residue class degrees of a prime ideal, with respect to an algebraic field extension;
• outline Minkowski's theorem;
• demonstrate the Four squares theorem with the help of Minkowski's theorem;
• in elementary cases apply Minkowski's bound for computation of the ideal class number;
• explain the concept "the geometry of numbers" according to Minkowski.

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. Discriminant of an algebraic number field. Lattices in Rⁿ and their quotient torus. Minkowski's theorem. Four squares theorem. Minkowski's bound. 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.