Galois groups and rational solutions of $p(X) = A$
| Autor: | Groenewald, G. J., Goosen, G., van Rensburg, D. B. Janse, Ran, A. C. M., van Straaten, M. |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $\mathbb{Q}.$ Comment: 17 pages |
| Databáze: | arXiv |
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