Hood College Department of Mathematics
Permanent URI for this collectionhttp://hdl.handle.net/11603/12980
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Item Connections Between Abstract Algebra and Polynomial Equations: The Legacy of Lagrange(2025-05-02) Porter, Elizabeth; Parson, James; Hood College Mathematics; Hood College Departmental HonorsThis paper investigates the evolution and theory of polynomial factorization, tracing a path from classical solutions of quadratics to modern techniques for analyzing quintic polynomials. We begin with foundational methods for factoring quadratics and cubics, including Cardano’s formula and its algebraic extensions. Moving into quartic equations, we compare the approaches of Ferrari and Descartes, and introduce Lagrange’s revolutionary perspective on root symmetries. This sets the stage for a deeper exploration into the structure of polynomials through Group Theory and Gröbner bases. While traditional methods fail to provide general solutions for the quintic, we demonstrate how modern algebraic tools and computational techniques allow us to investigate its structure through the lens of the symmetry.Item Enumerative Combinatorics and Postional Games(2017-04) Spessard, Clark; Whieldon, Gwyneth; Hood College Mathematics; Hood College Departmental HonorsGraph Avoidance games are a commonly studied type of combinatorial game. One class of graph avoidance games are Maker-Breaker games (Definition 3.4.1). In this thesis we prove weak solutions to Maker-Breaker games where Maker must construct a cycle, which we call Cycle Games (Definition 3.4.2), as well as provide methods to reduce the size of Cycle Game trees.