Talk by Federico Fornesi: Valuations, Q_p, and Their Extensions

On Friday, June 12, 2026, Lic. Federico Fornesi will give a talk as part of the IMERL Algebra Seminar.

Title: Valuations, Q_p, and Their Extensions

Time and Venue: Friday, June 12 at 11:30 AM – IMERL Seminar Room and via Zoom

Abstract: The theory of p-adic numbers emerges as a natural extension of the rational numbers, providing a powerful tool in various branches of number theory. One of the key results that justifies the study of p-adic numbers is Ostrowski’s Theorem, which classifies all absolute values on the rational numbers. In this way, p-adic numbers appear naturally when considering the notion of measuring distances in Q.

In this seminar, we will cover the basic definitions and concepts, followed by some algebraic properties such as Hensel’s Lemma, and analytical ones like Strassman’s Theorem. Subsequently, we will explore how to extend the absolute value from Q_p to its algebraic extensions, and discuss the study and classification of these extensions. We will conclude by discussing the algebraic closure of Q_p, defining C_p, and proving the Newton Polygon Theorem within this context.