Michel 1 Chapter 0 Welcome to Cyclotomic Polynomials. The following is an expository paper aimed at teaching readers the basic concepts leading to the irreducibility of the cyclotomic polynomials over Z. Carl Fredrick Gauss (1777 – 1855), an epoch-making mathematician, discovered the cyclotomic polynomials at the young age of 19. This discovery was first published as the seventh and final section of his monumental book on number theory
Disquisitiones Arithmeticae. Gauss was able to use these concepts to show exactly which polygons could be constructed by a compass and straight edge, as well as construct a 17sided polygon which, as far as the world knows, had never been constructed before. The scope of this paper, however, will not include polygon construction. After introducing and defining cyclotomic polynomials, we will look at some examples, and discuss the method of recursive calculation of these polynomials. Following this are four sections on several properties of the cyclotomic polynomials which contribute to their irreducibility over Z. In the final section of this paper we will discuss and prove their irreducibility. To understand the proofs the reader is recommended to have had a 300-level abstract algebra course or a solid background in algebra and number theory. Some proofs will only be mentioned in the index since the paper is limited on space. Now begins our journey.