Physical Sciences and Mathematics Commons^™

Galois 2-Extensions, Masoud Ataei Jaliseh

Electronic Thesis and Dissertation Repository

The inverse Galois problem is a major question in mathematics. For a given base field and a given finite group $G$, one would like to list all Galois extensions $L/F$ such that the Galois group of $L/F$ is $G$.

In this work we shall solve this problem for all fields $F$, and for group $G$ of unipotent $4 \times 4$ matrices over $\mathbb{F}_2$. We also list all $16$ $U_4 (\mathbb{F}_2)$-extensions of $\mathbb{Q}_2$. The importance of these results is that they answer the inverse Galois problem in some specific cases.

This is joint work with J\'an Min\'a\v{c} and Nguyen Duy T\^an.

Combinatorial Polynomial Identity Theory, Mayada Khalil Shahada

Electronic Thesis and Dissertation Repository

This dissertation consists of two parts. Part I examines certain Burnside-type conditions on the multiplicative semigroup of an (associative unital) algebra $A$.

A semigroup $S$ is called $n$-collapsing if, for every $a_1,\ldots, a_n \in S$, there exist functions $f\neq g$ on the set $\{1,2,\ldots,n\}$ such that \begin{center} $s_{f(1)}\cdots s_{f(n)} = s_{g(1)}\cdots s_{g(n)}$. \end{center} If $f$ and $g$ can be chosen independently of the choice of $s_1,\ldots,s_n$, then $S$ satisfies a semigroup identity. A semigroup $S$ is called $n$-rewritable if $f$ and $g$ can be taken to be permutations. Semple and Shalev extended Zelmanov's Fields Medal writing solution of the Restricted …

Algorithms To Compute Characteristic Classes, Martin Helmer

Electronic Thesis and Dissertation Repository

In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases).

We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and …

Combinatorial Techniques In The Galois Theory Of P-Extensions, Michael Rogelstad

Electronic Thesis and Dissertation Repository

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of p-extensions is an important step toward a solution. We illustrate several techniques for counting Galois p-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the F_p-dimension of an n-th …