Hamlet and Pfister forms (A tragedy in four acts)

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.