Extended Special Linear group $ESL_2(\mathbb{F})$ and square roots in matrix groups $SL_2(\mathbb{F})$, $SL_2(\mathbb{Z})$, $ESL_2(\mathbb{F})$, $GL_2(\mathbb{F}_p)$

Abstract

First time, we introduce Extended special linear group $ESL_2(F)$, which is generalization of matrix group $SL_2(F)$ over arbitrary field $F$. Extended special linear group $ESL_2(k)$, where $k$ is arbitrary perfect field, is storage of all square matrix roots from $ESL_2(k)$. The analytical formulas of roots of 2-nd, 3-rd, 4-th and $n$-th powers in $ SL_2(\mathbb{F}_p)$ are found by us. Also for roots in $ SL_2(\mathbb{Z})$, $ ESL_2(\mathbb{Z})$ and in $ SL_2({k})$ as well as in $ESL_2({k})$, where $k$ is arbitrary perfect field, is found by us. New linear group which is storage of square roots from $ SL_2{\mathbb{F}_p}$ is found and investigated by us. The criterion of roots existing for different classes of matrix -- simple and semisimple matrixes from $ SL_2({\mathbb{F}_p})$, $ SL_2({\mathbb{Z}})$ are established. The problems of square root from group element existing in $SL_2(F_p)$, $SL_2(F_p)$ and $GL_2(F_p)$ for arbitrary prime $p$ are solved in this paper. The similar goal of root finding was reached in the GM algorithm adjoining an $n$-th root of a generator results in a discrete group for group $SL(2,R)$, but we consider this question over finite field $F_p$. Over method gives answer about existing $\sqrt{ M^n}$ without exponenting $M$ to $n$-th power. We only use the trace of $M$ or only eigenvalues of $M$. In \cite{Amit} only the Anisotropic case of group $SL_1(Q)$, where $Q$ is a quaternion division algebra over $k$ was considered. Previously criterion to be square only for the case $F_p$ is a field of characteristics not equal 2 was considered. We solve this problem even for fields $F_2$ and $F_{2^n}$. The criterion to $g \in SL_2 (F_2)$ be square in $SL_2(F_2)$ was not found by them what was declared in a separate sentence in \cite{Amit}. We consider more general case consisting in whole group $G= SL_2(F_q)$. The structure of extended symplectic group is found.