We know that for any matrix $A\in GL_n(\mathbb C)$, and any natural number $m$ there exists a matrix $B\in GL_n(\mathbb C)$ such that $B^m=A$, can we extend this to any algebraically closed field with positive characteristic?
$nth$ root of matrices in characteristic $p$
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linear-algebra
matrices
1 Answers
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Not really. In characteristic $p$, consider a matrix $A$ with all eigenvalues $1$. Assume that $B^p=A$. Then the eigenvalues of $B$ are all $1$ ($x^p - 1=(x-1)^p$), and so $B-I$ is nilpotent. Assume now that $\dim B \le p$. Then $(B-I)^p=0$. However, $(B-I)^p= B^p- I^p = A-I$, and so $A=I$. Hence, a unipotent matrix not identity and of size at most $p$ does not have $p$-th roots.