A categorical approach to the theory of equations

Abstract

Listen

Translate article

We construct a small category whose objects are monic square-free polynomials with coefficients in a field F. For a monic, irreducible, and normal polynomial, Aut ( f) is the usual Galois group of f. We prove that there exists a unique topological group G such that the category of finite discrete G-sets is equivalent to the opposite of our category. We then replace categories by commutative rings and define the Burnside ring of a field, which has Burnside rings of finite groups as its building blocks. We next extend scalars to the rationals and explicity determine the algebra that results. We find all valuations of this algebra and prove that an irreducible polynomial is completely determined by its values under these valuations.