This paper is an investigation of the mathematics necessary to understand the Kronecker-Weber Theorem. Following an article by Greenberg, published in The American Mathematical Monthly in 1974, the presented proof does not use class field theory, as the most traditional treatments of the theorem do, but rather returns to more basic mathematics, like the original proofs of the theorem. This paper seeks to present the necessary mathematical background to understand the proof for a reader with a solid undergraduate background in abstract algebra. Its goal is to make what is usually an advanced topic in the study of algebraic number theory more accessible to advanced undergraduates and early graduate students, with a minimal amount of higher level number theory required. It also seeks to develop an appreciation for the power and elegance of this theorem and its role in mathematics, since it combines understanding in many branches — classical Galois theory, geometry, complex numbers, abelian groups, and number theory.
Level of Honors
Verser, Amber, "The Kronecker-Weber Theorem: An Exposition" (2013). Lawrence University Honors Projects. 52.