Physical Sciences and Mathematics Commons^™

Geometric Constructions, Origami, And Galois Theory, Julia Greene

Honors Theses

Geometric constructions using an unmarked straightedge and a compass have been studied for thousands of years. In these constructions, we can draw circles and lines starting with any two points, and we can create new points where they intersect. An n-gon is said to be constructible if can be constructed in a finite number of steps using these guidelines. We begin with constructions of several n-gons, and examine the field theory behind geometric constructions. Galois theory then provides a precise classification of which n-gons are constructible and which are not. Next is an exploration of origami construction, which examines a …

Galois Theory And The Quintic Equation, Yunye Jiang

Honors Theses

Most students know the quadratic formula for the solution of the general quadratic polynomial in terms of its coefficients. There are also similar formulas for solutions of the general cubic and quartic polynomials. In these three cases, the roots can be expressed in terms of the coefficients using only basic algebra and radicals. We then say that the general quadratic, cubic, and quartic polynomials are solvable by radicals. The question then becomes: Is the general quintic polynomial solvable by radicals? Abel was the first to prove that it is not. In turn, Galois provided a general method of determining when …

Parametric Polynomials For Small Galois Groups, Claire Huang

Honors Theses

Galois theory, named after French mathematician Evariste Galois in 19th-century, is an important part of abstract algebra. It brings together many different branches of mathematics by providing connections among fields, polynomials, and groups.

Specifically, Galois theory allows us to attach a finite field extension with a finite group. We call such a group the Galois group of the finite field extension. A typical way to attain a finite field extension to compute the splitting field of some polynomial. So we can always start with a polynomial and find the finite group associate to the field extension on its splitting field. …

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.

On Dedekind’S “Über Die Permutationen Des Körpers Aller Algebraischen Zahlen", Joseph Jp Arsenault Jr

Electronic Theses and Dissertations

We provide an analytic read-through of Richard Dedekind's 1901 article “Über die Permutationen des Körpers aller algebraischen Zahlen," describing the principal results concerning infinite Galois theory from both Dedekind's point of view and a modern perspective, noting an apparently uncorrected error in the supplement to the article in the Collected Works. As there is no published English-language translation of the article, we provide an annotated original translation.

Irreducibility And Galois Properties Of Lifts Of Supersingular Polynomials, Rylan Jacob Gajek-Leonard

Senior Projects Spring 2015

It has recently been shown that a rational specialization of Jacobi polynomials, when reduced modulo a prime number p, has roots which coincide with the supersingular j- invariants of elliptic curves in characteristic p. These supersingular lifts are conjectured to be irreducible with maximal Galois groups. Using the theory of p-adic Newton Polygons, we provide a new infinite class of irreducibility and, assuming a conjecture of Hardy and Littlewood, give strong evidence for their Galois groups being as large as possible.

Galois Representations From Non-Torsion Points On Elliptic Curves, Matthew Phillip Hughes

Senior Projects Spring 2013

Working from well-known results regarding l-adic Galois representations attached to elliptic curves arising from successive preimages of the identity, we consider a natural deformation. Given a non-zero point P on a curve, we investigate the Galois action on the splitting fields of preimages of P under multiplication-by-l maps. We give a group-theoretic structure theorem for the corresponding Galois group, and state a conjecture regarding composita of two such splitting fields.