On the Applications of Cyclotomic Fields in Introductory Number Theory

Details On the Applications of Cyclotomic Fields in Introductory Number Theory On the Applications of Cyclotomic Fields in Introductory Number Theory

2011-10-20

arxiv.org/abs/1110.4445v2

Mathematics

Number Theory

Scientific paper

In this essay, we see how cyclotomic fields can lead to elegant proofs of number theoretical concepts. We will prove some elementary properties of prime cyclotomic fields (a cyclotomic field obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime), and use them to prove the laws of Quadratic and Cubic Reciprocity. We will also explore the applications of cyclotomic fields in certain forms of Diophantine equations. We namely develop the notion of primary units in a cyclotomic field and show how they lead to a proof of a special case of Fermat's Last Theorem. We namely modernize Dirichlet's solution to Pell's Equation.

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