Exercise Sessions in Foundations of Number Theory
Tuesdays 8-10 (Room 0.008), winter term 2023/2024, Uni Bonn
Anouncements
Exercise sessions start at 8:30!
Solutions to the exercise sheets
Warning: These are my preperations for the exercise sessions. There might be
errors, some solutions might even be completely wrong. If you think you found
an error or have questions regarding the solutions, I strongly encourage you to
tell me about it :)
- Sheet 1 (mostly) results on algebraic integers.
- Sheet 2 Computations with norm and trace, a integral extension of the integers with infinitely many units, all finite free modules over integral domains are submodules of a free module.
- Sheet 3 More units of rings of integers, quadratic extensions, cubic extensions, quadratic extensions of quadratic extensions (and fun facts about the flag of Nepal!).
- Sheet 4 Discriminants of quadratic number fields, further calculations with special cubic number fields, first facts about Dedekind domains.
- Sheet 5 Calculations of integer rings are completed (but are painful), inverse of ideals, some algebraic extensions of the integers are not Dedekind.
- Sheet 6 Rings of integers have infinitely many primes,
a certain polynomial only takes on prime values (at least if you don’t check for large inputs), ideals behave well with respect to field automorphisms, dedekind rings with finitely many ideals are principal.
- Sheet 7 Number of Ideals with given norm in quadratic
and cubic extensions, a non-monogenic number field.
- Sheet 8 Ramification of primes, Dedekind coefficients are multiplicative.
- Sheet 9 The legendre symbol in the context of ramification of primes, discriminants of p-th roots of unity.
- Sheet 10 Fun proof of the infinitude of primes that are 1 mod 4, calculations with discriminants, direct calculation of class groups (without Minkowski’s bound).
- Sheet 11 Every prime can be written as a sum of four squares, calculation of class numbers of certain quadratic fields, some explicit verification of Dirichlet’s unit theorem.
- Sheet 12 Every ideal has trivial ideal class in some extension, some subgroup of the units, an exercise that wants you to understand the proof of Dirichlet’s unit theorem and a first application of analytic tools.
- Sheet 13 Something something principal ideals, Dedekind-zeta functions, analysis. Update: Fixed a mistake in exercise 1.
- Sheet 14 Some analytic number theory, a neat formula for decomposing Dedekind zeta functions of abelian number fields, a particular element is a unit.
Max von Consbruch, email: mvconsbruch(at)uni-bonn(dot)de