This is the senior thesis I worked on in University of Washington, under the supervision of Prof. Ralph Greenberg. We intended to classify the torsion-free part of the unit group of a totally complex normal extension with Galois group S3 as a Z[S3]-module. We obtained a way to recognize the module structure based on the class numbers of the field extensions. Conversely we also showed that some ...

Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin’s elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a c...

Given a symmetric polynomial 8(x, y) over a perfect field k of characteristic zero, the Galois graph G(8) is defined by taking the algebraic closure k̄ as the vertex set and adjacencies corresponding to the zeroes of 8(x, y). Some graph properties of G(8), such as lengths of walks, distances and cycles are described in terms of 8. Symmetry is also considered, relating the Galois group Gal(k̄/k) t...

Let En be the n-th Lubin-Tate spectrum at a prime p. There is a commutative S-algebra E n whose coefficients are built from the coefficients of En and contain all roots of unity whose order is not divisible by p. For odd primes p we show that E n does not have any non-trivial connected finite Galois extensions and is thus separably closed in the sense of Rognes. At the prime 2 we prove that the...

Fix K a p-adic field and denote by GK its absolute Galois group. Let K∞ be the extension of K obtained by adding p-th roots of a fixed uniformizer, and G∞ ⊂ GK its absolute Galois group. In this article, we define a class of p-adic torsion representations of G∞, named quasi-semi-stable. We prove that these representations are “explicitly” described by a certain category of linear algebra object...

This paper presents works on reducible polynomials in Galois theory. It gives a fundamental theorem about the Galois ideals associated with a product of groups. These results are used for the practical determination of Galois groups of polynomials of degrees up to 7. : : :). The following tools are used: resolvents (relative or absolute), p-adic methods, Grr obner bases or block systems. In gen...

It is shown that every quantum principal bundle with a compact structure group is a Hopf-Galois extension. This property naturally extends to the level of general differential structures, so that every differential calculus over a quantum principal bundle with a compact structure group is a graded-differential variant of the Hopf-Galois extension.

Some applications are displayed: these include a Galois cohomological descent spectral sequence for the étale K–theory of a scheme (where the Galois group is the Grothendieck fundamental group), and the Morel–Voevodsky description of Thomason–Trobaugh K–theory as Nisnevich K–theory in nonnegative degrees. The is also a spectrum-level description of Voevodsky’s periodicity operator for Nisnevich...

Let p be a prime number and M a quadratic number field, M , Q( √ p) if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D2p and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least pd.