Galois groups and rational solutions of $p(X) = A$

G. J. Groenewald; G. Goosen; D. B. Janse van Rensburg; A. C. M. Ran; M. van Straaten

Galois groups and rational solutions of $p(X) = A$

Number Theory 2023-07-13 v1 Functional Analysis

Authors: G. J. Groenewald , G. Goosen , D. B. Janse van Rensburg , A. C. M. Ran , M. van Straaten

Abstract

We extend Theorem 1 of R. Reams, A Galois approach to m-th roots of matrices with rational entries, LAA 258 (1997), 187-194. Let $p(\lambda)$ be any polynomial over $\mathbb{Q}$ and let $A\in M_n(\mathbb{Q})$ have irreducible characteristic polynomial $f(\lambda)$ with degree n. We provide necessary and sufficient conditions for the existence of a solution $X\in M_n(\mathbb{Q})$ of the polynomial matrix equation $p(X) = A.$ Specifically, we find necessary and sufficient conditions for $f(p(\lambda))$ to have a factor of degree $n$ over $\mathbb{Q}.$

Keywords

galois groups rational solutions polynomial matrix equation characteristic polynomial irreducible polynomial

Cite

@article{arxiv.2307.06303,
  title  = {Galois groups and rational solutions of $p(X) = A$},
  author = {G. J. Groenewald and G. Goosen and D. B. Janse van Rensburg and A. C. M. Ran and M. van Straaten},
  journal= {arXiv preprint arXiv:2307.06303},
  year   = {2023}
}

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17 pages

Galois groups and rational solutions of $p(X) = A$

Abstract

Keywords

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