# The images of non-commutative polynomials evaluated on the Quaternion algebra

## TL;DR

This paper proves a conjecture about the images of non-commutative polynomials evaluated on quaternion algebras, showing they form vector spaces, extending prior matrix algebra results.

## Contribution

It establishes the conjecture for quaternion algebras, a significant extension of Kaplansky's conjecture from matrix algebras.

## Key findings

- The image of multilinear polynomials on quaternion algebras is a vector space.
- The result confirms the conjecture in the context of quaternion algebras.
- It broadens understanding of polynomial evaluations in non-commutative algebraic structures.

## Abstract

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector space. In this paper we settle the analogous conjecture for a quaternion algebra.