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Full-Text Articles in Algebra
Galois Theory And The Quintic Equation, Yunye Jiang
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 …
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
An Algebraic Approach To Number Theory Using Unique Factorization, Mark Sullivan
Honors Theses
Though it may seem non-intuitive, abstract algebra is often useful in the study of number theory. In this thesis, we explore some uses of abstract algebra to prove number theoretic statements. We begin by examining the structure of unique factorization domains in general. Then we introduce number fields and their rings of algebraic integers, whose structures have characteristics that are analogous to some of those of the rational numbers and the rational integers. Next we discuss quadratic fields, a special case of number fields that have important applications to number theoretic problems. We will use the structures that we introduce …