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Full-Text Articles in Physical Sciences and Mathematics
Genus Bounds For Some Dynatomic Modular Curves, Andrew W. Herring
Genus Bounds For Some Dynatomic Modular Curves, Andrew W. Herring
Electronic Thesis and Dissertation Repository
We prove that for every $n \ge 10$ there are at most finitely many values $c \in \mathbb{Q} $ such that the quadratic polynomial $x^2 + c$ has a point $\alpha \in \mathbb{Q} $ of period $n$. We achieve this by proving that for these values of $n$, every $n$-th dynatomic modular curve has genus at least two.
Algebraic Number Theory And Simplest Cubic Fields, Jianing Yang
Algebraic Number Theory And Simplest Cubic Fields, Jianing Yang
Honors Theses
The motivation behind this paper lies in understanding the meaning of integrality in general number fields. I present some important definitions and results in algebraic number theory, as well as theorems and their proofs on cyclic cubic fields. In particular, I discuss my understanding of Daniel Shanks' paper on the simplest cubic fields and their class numbers.
Combinatorial Techniques In The Galois Theory Of P-Extensions, Michael Rogelstad
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 Fp-dimension of an n-th …
Algebraic And Combinatorial Properties Of Schur Rings Over Cyclic Groups, Andrew F. Misseldine
Algebraic And Combinatorial Properties Of Schur Rings Over Cyclic Groups, Andrew F. Misseldine
Theses and Dissertations
In this dissertation, we explore the nature of Schur rings over finite cyclic groups, both algebraically and combinatorially. We provide a survey of many fundamental properties and constructions of Schur rings over arbitrary finite groups. After specializing to the case of cyclic groups, we provide an extensive treatment of the idempotents of Schur rings and a description for the complete set of primitive idempotents. We also use Galois theory to provide a classification theorem of Schur rings over cyclic groups similar to a theorem of Leung and Man and use this classification to provide a formula for the number of …
Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre
Factorization Of Primes Primes Primes: Elements Ideals And In Extensions, Peter J. Bonventre
Honors Theses
It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored uniquely into primes. However, if K is a finite extension of the rational numbers, and OK its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into prime ideals. This result allows us to extend many properties of the integers to these rings. If we a finite extension L of K and OL of OK , we find that …
The Solvability Of Polynomials By Radicals: A Search For Unsolvable And Solvable Quintic Examples, Robert Lewis Beyronneau
The Solvability Of Polynomials By Radicals: A Search For Unsolvable And Solvable Quintic Examples, Robert Lewis Beyronneau
Theses Digitization Project
This project centers around finding specific examples of quintic polynomials that were and were not solvable. This helped to devise a method for finding examples of solvable and unsolvable quintics.