Hamlet and Pfister forms (A tragedy in four acts)

TL;DR

This paper explores the mathematical properties of Pfister forms, a significant class of quadratic forms, and presents a new proof technique, while also drawing a poetic analogy to Shakespeare's Hamlet to illustrate the forms' mysterious and influential nature.

Contribution

It introduces a novel proof method for Pfister forms using field multiplication and creatively links the mathematical concept to the tragedy of Hamlet.

Findings

New proof technique for Pfister forms using field elements

Pfister forms' properties related to polynomial similarity factors

Poetic analogy linking mathematical forms to Hamlet's tragedy

Abstract

In the mid-1960s A. Pfister discovered extraordinary, strongly multiplicative forms which are now called Pfister forms. From that time on, these forms played a dominant role in the theory of quadratic forms. One of the key properties of a Pfister form q is that q extended to a suitable transcendental extension, has the polynomial q as its similarity factor. Pfister's original proof used clever matrix calculations. Here we show that the desired isometry is induced by the multiplication of a suitable field element. We further consider the surprising possibility that Pfister's forms were already known by Hamlet, Rosencrantz and Guildenstern, and that they in fact led to a terrible tragedy which is yet filled with a haunting beauty and mystery that can still inspire us to this day.