On the Intermediate Value Theorem over a Valued Field

TL;DR

This paper extends the intermediate value theorem to polynomials and power series over valued fields with specific properties, using valuation theory and Hensel's Lemma to explore valuation behavior.

Contribution

It establishes the intermediate value theorem in the context of valued fields with divisible valuation groups and infinite residue fields, advancing valuation theory.

Findings

Proves the intermediate value theorem for polynomials over certain valued fields.

Derives new results on valuation behavior using Hensel's Lemma.

Provides insights into the structure of valued fields with divisible valuation groups.

Abstract

The paper proves the intermediate value theorem for polynomials and power series over a valued field with divisible valuation group and infinite residue field. Some further results on the behaviour of the valuation are obtained using Hensel's Lemma.