# Number Theory Seminar

Presented by: Keith Johnson

Title: “Rational Polynomials which Preserve the Ring of 2x2 Matrices with Integer Coefficients”

Abstract

If $M_2(\mathbb{Z})$ denotes the ring of $2\times 2$ matrices with coefficients in $\mathbb{Z}$ then there are polynomials in $p(x)\in\mathbb{Q}[x]$ for which $p(M)\in M_2(\mathbb{Z})$ if $M\in M_2(\mathbb{Z})$ without $p(x)$ having all integer coefficients itself. $(x^6+x^5+x^3+x^2)/2$ is an example. Describing all polynomials in $\mathbb{Q}[x]$ with this property is an open problem and this talk will describe how much is known so far.

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#### Location

#319 Colloquium Room, Chase Building