Subalgebras of Matrix Algebras Generated by Companion Matrices

TL;DR

This paper characterizes when subalgebras generated by companion matrices of polynomials over integers or rings are of finite index in full matrix algebras, providing explicit index formulas and conditions for equality.

Contribution

It establishes necessary and sufficient conditions for subalgebras generated by companion matrices to be of finite index and computes the exact index using resultants.

Findings

Conditions for subalgebras to have finite index in M_n(Z)

Explicit formula for the index in terms of the resultant

Criteria for when the subalgebra equals the entire matrix algebra over a ring

Abstract

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative ring with identity we determine when $R< C,D>=M_n(R)$, in which case a presentation for $M_n(R)$ in terms of $C$ and $D$ is given.