CU Graduate Mathematics Society

Kronecker-Weber Theorem---Edward Sanders

The Kronecker-Weber Theorem states that any abelian field extension of the rationals is a subfield of a cyclotomic field extension of the rationals. The Global Kronecker-Weber Theorem can be derived from the Local Kronecker-Weber Theorem. The theorem that derives this result uses ideas that relate the Galois groups field extensions over the rationals compared to the p-adics such as decomposition and Inertia groups. I will also illustrate a method of proof for the Local case.

Kummer Theory and the Hilbert Symbol---Martin Orr
Kummer theory provides a description of extensions obtained by adjoining nth roots to a field K, where K already contains all nth roots of unity.  As well as showing that these are precisely the abelian extensions of exponent n, Kummer theory describes the character group of the Galois group of such extensions.  This talk will require basic knowledge of Galois theory and nothing else.

Infinite Galois Theory--- Barinder Banwait

The theory of finite Galois Extensions is well-known, but the results do
not generalise to the infinite case without some extra work. In this talk I define the Krull Topology for infinite Galois Extensions, state the
Fundamental theorem, and possibly state a theorem by Artin and Schreier which ensures the existence of a large class of infinite Galois extensions. I then compute some examples, and end by considering the "Inverse problem of Galois Theory"; given a finite group G, does there exist a finite extension L/K whose Galois group is G? I consider K = Qp, where the answer is No, and K = Qab, where the answer is Yes, provided a conjecture of Shafaravich holds.

Dirichlet's Theorem on arithmetic progressions---Konrad Dabrowski

Dirichlet's Theorem says that if a and d are coprime then there are infinitely many primes of the form a+nd. At some point, you may have seen this proved for some small values of a and d. I will present only part of the proof of the theorem and will then show that this is sufficient to prove it for some values of d with very little effort.