Minimal and characteristic polynomials of symmetric matrices in\n characteristic two

2021/06/15 by Grégory Berhuy, Berhuy, Grégory
Engineering · Mathematics · #11C20 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #\ 11E39 #\ 15A15 #\ 15A18 #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2106.10239

openalex publication_date 2021/06/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a field of characteristic two. We prove that a non constant monic\npolynomial f\∈ k[X] of degree n is the minimal/characteristic polynomial\nof a symmetric matrix with entries in k if and only if it is not the product\nof pairwise distinct inseparable irreducible polynomials. In this case, we\nprove that f is the minimal polynomial of a symmetric matrix of size n. We\nalso prove that any element \α\∈ k_alg of degree n\≥ 1 is the\neigenvalue of a symmetrix matrix of size n or n+1, the first case happening\nif and only if the minimal polynomial of \α is not the product of\npairwise distinct inseparable irreducible polynomials.\n

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