# 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.

#### Category

Lectures, Seminars

#### Time

Starts:

Ends:

#### Location

Chase Building, Room 319

#### Cost

Free

#### Contact

Lin Jiu