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Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But what about ${\rm SL}_n(\mathbb K)$? Again, the answer is negative if $\mathbb K$ has finite characteristic, and again we may assume that (i) our matrices are in Jordan normal form and (ii) $\mathbb K$ is the complex field, thanks to the Lefschetz principle and the fact that, by Laplace's formula, the determinant of an $n$-by-$n$ "formal matrix" can be expressed as a wff in the first-order language $\mathcal L = (+, \cdot, -, 0, 1)$ of (the theory of) rings. Thus, let us consider a Jordan matrix of size $n \times n$, say $J = {\rm diag}(J_1, \ldots, J_m)$, where $J_i$ is a Jordan block of size $k_i \times k_i$. If $\det(J) = 1$ and $p \nmid \gcd(k_1, \ldots, k_m)$, it is not difficult to prove that there exists $A \in {\rm SL}_n(\mathbb C)$ such that $A^p = J$.

Question 1. What about the other cases?

The problem should be well-known and I feel that the answer is in the negative, but so far I couldn't either get a reference or find a counterexample by myself. EDIT 2. The $2$-by-$2$ Jordan block with eigenvalue $-1$ is a counterexample, and in some sense it is the only one possible for $n = 2$ (see the comments below). Thus, it seems natural to ask the following:

Question 2. (i) Is there any "explicit characterization" of those matrices in ${\rm SL}_n(\mathbb C)$ which fail to have a $p$-th root for each prime $p$? (ii) In particular, is the set of these matrices the union of a finite number of conjugacy classes of ${\rm SL}_n(\mathbb C)$?

Let me rely on your common sense for the actual meaning to give to the expression "explicit characterization". Last but not least:

Question 3. Does anyone know where to find a reference for this kind of questions (concerned with the divisibility of specific subgroups of ${\rm GL}_n(\mathbb K)$ when $\mathbb K$ is an algebraically closed field, either in characteristic zero or not)?

Thanks in advance for any help.

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But what about ${\rm SL}_n(\mathbb K)$? Again, the answer is negative if $\mathbb K$ has finite characteristic, and again we may assume that (i) our matrices are in Jordan normal form and (ii) $\mathbb K$ is the complex field, thanks to the Lefschetz principle and the fact that, by Laplace's formula, the determinant of an $n$-by-$n$ "formal matrix" can be expressed as a wff in the first-order language $\mathcal L = (+, \cdot, -, 0, 1)$ of (the theory of) rings. Thus, let us consider a Jordan matrix of size $n \times n$, say $J = {\rm diag}(J_1, \ldots, J_m)$, where $J_i$ is a Jordan block of size $k_i \times k_i$. If $\det(J) = 1$ and $p \nmid \gcd(k_1, \ldots, k_m)$, it is not difficult to prove that there exists $A \in {\rm SL}_n(\mathbb C)$ such that $A^p = J$.

Question 1. What about the other cases?

The problem should be well-known and I feel that the answer is in the negative, but so far I couldn't either get a reference or find a counterexample by myself. EDIT 2. The $2$-by-$2$ Jordan block with eigenvalue $-1$ is a counterexample, and in some sense it is the only one possible for $n = 2$ (see the comments below). Thus, it seems natural to ask the following:

Question 2. (i) Is there any "explicit characterization" of those matrices in ${\rm SL}_n(\mathbb C)$ which fail to have a $p$-th root for each prime $p$? (ii) In particular, is the set of these matrices the union of a finite number of conjugacy classes of ${\rm SL}_n(\mathbb C)$?

Let me rely on your common sense for the actual meaning to give to the expression "explicit characterization". Last but not least:

Question 3. Does anyone know where to find a reference for this kind of questions (concerned with the divisibility of specific subgroups of ${\rm GL}_n(\mathbb K)$ when $\mathbb K$ is an algebraically closed field, either in characteristic zero or not)?

Thanks in advance for any help.

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But what about ${\rm SL}_n(\mathbb K)$? Again, the answer is negative if $\mathbb K$ has finite characteristic, and again we may assume that (i) our matrices are in Jordan normal form and (ii) $\mathbb K$ is the complex field, thanks to the Lefschetz principle and the fact that, by Laplace's formula, the determinant of an $n$-by-$n$ "formal matrix" can be expressed as a wff in the first-order language $\mathcal L = (+, \cdot, -, 0, 1)$ of (the theory of) rings. Thus, let us consider a Jordan matrix of size $n \times n$, say $J = {\rm diag}(J_1, \ldots, J_m)$, where $J_i$ is a Jordan block of size $k_i \times k_i$. If $\det(J) = 1$ and $p \nmid \gcd(k_1, \ldots, k_m)$, it is not difficult to prove that there exists $A \in {\rm SL}_n(\mathbb C)$ such that $A^p = J$.

Question 1. What about the other cases?

The problem should be well-known and I feel that the answer is in the negative, but so far I couldn't either get a reference or find a counterexample by myself. EDIT. The $2$-by-$2$ Jordan block with eigenvalue $-1$ is a counterexample (see the comments below).

Question 2. Does anyone know where to find a reference for this kind of questions (namely, the divisibility of special subgroups of ${\rm GL}_n(\mathbb K)$ when $\mathbb K$ is an algebraically closed field of zero characteristic)?

Thanks in advance for any help.

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But what about ${\rm SL}_n(\mathbb K)$? Again, the answer is negative if $\mathbb K$ has finite characteristic, and again we may assume that (i) our matrices are in Jordan normal form and (ii) $\mathbb K$ is the complex field, thanks to the Lefschetz principle and the fact that, by Laplace's formula, the determinant of an $n$-by-$n$ "formal matrix" can be expressed as a wff in the first-order language $\mathcal L = (+, \cdot, -, 0, 1)$ of (the theory of) rings. Thus, let us consider a Jordan matrix of size $n \times n$, say $J = {\rm diag}(J_1, \ldots, J_m)$, where $J_i$ is a Jordan block of size $k_i \times k_i$. If $\det(J) = 1$ and $p \nmid \gcd(k_1, \ldots, k_m)$, it is not difficult to prove that there exists $A \in {\rm SL}_n(\mathbb C)$ such that $A^p = J$.

Question 1. What about the other cases?

The problem should be well-known and I feel that the answer is in the negative, but so far I couldn't either get a reference or find a counterexample by myself. EDIT 2. The $2$-by-$2$ Jordan block with eigenvalue $-1$ is a counterexample, and in some sense it is the only one possible for $n = 2$ (see the comments below). Thus, it seems natural to ask the following:

Question 2. (i) Is there any "explicit characterization" of those matrices in ${\rm SL}_n(\mathbb C)$ which fail to have a $p$-th root for each prime $p$? (ii) In particular, is the set of these matrices the union of a finite number of conjugacy classes of ${\rm SL}_n(\mathbb C)$?

Let me rely on your common sense for the actual meaning to give to the expression "explicit characterization". Last but not least:

Question 3. Does anyone know where to find a reference for this kind of questions (concerned with the divisibility of specific subgroups of ${\rm GL}_n(\mathbb K)$ when $\mathbb K$ is an algebraically closed field, either in characteristic zero or not)?

Thanks in advance for any help.

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But what about ${\rm SL}_n(\mathbb K)$? Again, the answer is negative if $\mathbb K$ has finite characteristic, and again we may assume that (i) our matrices are in Jordan normal form and (ii) $\mathbb K$ is the complex field, thanks to the Lefschetz principle and the fact that, by Laplace's formula, the determinant of an $n$-by-$n$ "formal matrix" can be expressed as a wff in the first-order language $\mathcal L = (+, \cdot, -, 0, 1)$ of (the theory of) rings. Thus, let us consider a Jordan matrix of size $n \times n$, say $J = {\rm diag}(J_1, \ldots, J_m)$, where $J_i$ is a Jordan block of size $k_i \times k_i$. If $\det(J) = 1$ and $p \nmid \gcd(k_1, \ldots, k_m)$, it is not difficult to prove that there exists $A \in {\rm SL}_n(\mathbb C)$ such that $A^p = J$. In particular, ${\rm SL}_2(\mathbb C)$ is divisible.

Question 1. What about the other cases?

The problem should be well-known and I feel that the answer is in the negative, but so far I couldn't either get a reference or find a counterexample by myself.

Question 2. Does anyone know where to find either/both of them?

Thanks in advance for any help.

Let $n$ be a positive integer, $p$ a (positive rational) prime, and $\mathbb K$ an algebraically closed field. If ${\rm char}(\mathbb K) = 0$ then ${\rm GL}_n(\mathbb K)$ is divisible (see here). But what about ${\rm SL}_n(\mathbb K)$? Again, the answer is negative if $\mathbb K$ has finite characteristic, and again we may assume that (i) our matrices are in Jordan normal form and (ii) $\mathbb K$ is the complex field, thanks to the Lefschetz principle and the fact that, by Laplace's formula, the determinant of an $n$-by-$n$ "formal matrix" can be expressed as a wff in the first-order language $\mathcal L = (+, \cdot, -, 0, 1)$ of (the theory of) rings. Thus, let us consider a Jordan matrix of size $n \times n$, say $J = {\rm diag}(J_1, \ldots, J_m)$, where $J_i$ is a Jordan block of size $k_i \times k_i$. If $\det(J) = 1$ and $p \nmid \gcd(k_1, \ldots, k_m)$, it is not difficult to prove that there exists $A \in {\rm SL}_n(\mathbb C)$ such that $A^p = J$.

Question 1. What about the other cases?

The problem should be well-known and I feel that the answer is in the negative, but so far I couldn't either get a reference or find a counterexample by myself. EDIT. The $2$-by-$2$ Jordan block with eigenvalue $-1$ is a counterexample (see the comments below).

Question 2. Does anyone know where to find a reference for this kind of questions (namely, the divisibility of special subgroups of ${\rm GL}_n(\mathbb K)$ when $\mathbb K$ is an algebraically closed field of zero characteristic)?

Thanks in advance for any help.