Valuations on Structures More General Than Fields

Abstract

Valuation theory is an important area of investigation in algebra, with applications in algebraic geometry and number theory. In 1957, M. Krasner introduced hyperfields, which are field-like objects with a multivalued addition, to describe some structures arising naturally from valued fields. We wish to discuss the possibility of generalising the notion of valuation to the multivalued setting and the potential that this higher point of view has in the understanding of classical valuation theory. We will see that a valuation on a field K is nothing but a homomorphism of hyperfields from K onto a special type of hyperfield, which we call a (generalised) tropical hyperfield.

Keywords

• hyperfield
• valuation
• valued field
• tropical hyperfield