Here are some proposals for student projects. All of them have some more or less strong relation to the lecture on Commutative Algebra.
Please contact me if you are interested in one of these subjects.
Galois theory has many famous applications, like the (non-)solvability of polynomial equations by taking successive roots, the impossibility of trisecting an angle with ruler and compass, etc. The main theorem of Galois theory establishes a bijection between the subfields of a finite Galois extension and subgroups of the automorphism group of this extension. An infinite version of this theorem also holds. For this, one has to put a topology on the automorphism group (it becomes a profinite group) and has to restrict to closed subgroups. The aim is to present profinite groups and this theorem.
References:
Neukirch: Algebraische Zahlentheorie (or: Algebraic Number Theory), Chapters IV.1 and IV.2.
The real numbers are the completion of the rationals with respect to the `usual' absolute value.
But, there are more absolute values on the rationals, for each prime p there is the so-called
p-absolute value. It behaves in a `non-archimedean way', i.e. differently from what one might
be used to from the `usual archimedean' absolute value.
Completing the rationals with respect to the p-absolute value gives a very interesting field,
called the field of p-adic numbers. One can do analysis on this field, e.g. study criteria for
the convergence of series, etc. One finds that the theory is actually much simpler than what
one is used to from the reals.
References:
Koblitz: p-adic numbers, p-adic analysis and zeta-functions, Chapter 1.
Schikhof: Ultrametric Calculus.
Fresnel, van der Put: Rigid analytic geometry and its applications, Chapter 1.
The main structure theorem of finitely generated modules over the Iwasawa algebra is similar to the structure theorem of finitely generated modules over a principal ideal domain. This subject is very advanced, since one needs completions and p-adic numbers at several points.
References:
Washington: Introduction to Cyclotomic Fields
Lang: Cyclotomic Fields I and II, 5.1-5.3
Already Euclid showed the existence of infinitely many primes. Although prime numbers seem to appear in quite a random way, there is a lot one can say about their distribution: the function that counts the primes up to a constant x behaves asymptotically like x/log(x). The aim is to prove this statement. Methods are mostly from complex analysis, and thus the subject only has a weak link with commutative algebra.
References:
This is a very well covered subject and so there is a wide range of references to choose from
The aim is to explain the theory and implement factorisation algorithms for polynomials over finite fields (and the integers).
References:
Cohen: A Course in Computational Algebraic Number Theory, 3.4 and 3.5
Modular forms play a very important role in current number theory. They are objects
coming from 19th century complex analysis. Modular forms can be
represented as Fourier series; often their Fourier coefficients are highly interesting
number theoretic functions, like the sum of all divisors of a natural number n.
The aim is to introduce modular forms for the full modular group and to present
some of their basic properties.
References:
Serre: Cours d'aritmétique / A Course in Arithmetic
Last modification: 12 October 2011.