特征为2的对称矩阵的极小特征多项式
Minimal and characteristic polynomials of symmetric matrices in characteristic two
摘要：
设$k$是特征为2的域。证明了$n$次的非常数幺正多项式$f\in k[X]$是元素在$k$中的对称矩阵的最小特征多项式当且仅当它不是两两不同的不可约不可约多项式的乘积.在这种情况下，我们证明$f$是大小为$n$的对称矩阵的最小多项式。我们还证明了任何元素$α\in k_{alg}$的程度$n\geq 1$是一个对称矩阵的大小$n$或$n+1$的特征值，第一种情况发生的当且仅当最小多项式的$α$不是产品的两两不同不可分的不可约多项式。
英文摘要：
Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of pairwise distinct inseparable irreducible polynomials. In this case, we prove that $f$ is the minimal polynomial of a symmetric matrix of size $n$. We also prove that any element $α\in k_{alg}$ of degree $n\geq 1$ is the eigenvalue of a symmetrix matrix of size $n$ or $n+1$, the first case happening if and only if the minimal polynomial of $α$ is not the product of pairwise distinct inseparable irreducible polynomials.
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