On algebraic extensions and algebraic closures of superfields

TL;DR

This paper extends classical algebraic extension theory to superfields, establishing the existence of algebraic closures and quantifier elimination for infinite algebraically closed superfields, thus broadening algebraic structures.

Contribution

It introduces the theory of algebraic extensions for superfields, proving the existence of algebraic closures and quantifier elimination in this new context.

Findings

Every superfield has a unique algebraic closure up to isomorphism.

Infinite algebraically closed superfields admit quantifier elimination.

The theory generalizes classical algebraic concepts to superfields.

Abstract

Building over recent results, we expand the basic theory of algebraic extensions to the realm of superfields -a field with multivalued sum and product-, showing that every superfield has a (unique up to isomorphism) strong algebraic extension to a superfield that is algebraically closed. Moreover we show that every infinite algebraically closed superfield admits quantifier elimination procedure.