Monogenic Cyclotomic compositions

Cyclotomic Polynomials

If n is a positive integer, then the n cyclotomic polynomial is defined as the unique monic polynomial having exactly the primitive n roots of unity as its zeros. In this paper we start off by examining some of the properties of cyclotomic polynomials; specifically focusing on their irreducibility and how they relate to primes. After that we explore some applications of these polynomials, inclu...

Cyclotomic Extensions

For any field K, a field K(ζn) where ζn is a root of unity (of order n) is called a cyclotomic extension of K. The term cyclotomic means circle-dividing, and comes from the fact that the nth roots of unity divide a circle into arcs of equal length. We will see that the extensions K(ζn)/K have abelian Galois groups and we will look in particular at cyclotomic extensions of Q and finite fields. T...

Classification of Monogenic Ternary Semigroups

The aim of this paper is to classify all monogenic ternary semigroups, up to isomorphism. We divide them to two groups: finite and infinite. We show that every infinite monogenic ternary semigroup is isomorphic to the ternary semigroup O, the odd positive integers with ordinary addition. Then we prove that all finite monogenic ternary semigroups with the same index...

Cyclotomic Fields

Cyclotomic fields are an interesting laboratory for algebraic number theory because they are connected to fundamental problems Fermat’s Last Theorem for example and also have relatively simple algebraic properties that makes them an excellent laboratory for results in algebraic number theory. I will assume that you are familiar with basic algebraic number theory. Namely, the unique factorizatio...

Cyclotomic Polynomials