Element structure of general linear group of degree two

You might be interested in:

• Element structure of general linear group of degree two over a field: Discussion for $GL(2,K)$, $K$ any field. The discussion involves a large number of cases based on the nature of $K$.
• Element structure of general linear group of degree two over a finite field: Discussion for $GL(2,q)=GL(2,\mathbb {F} _{q})$, $q$ a prime power. This is a more manageable and concrete discussion than the generic case.
• Element structure of general linear group of degree two over a finite discrete valuation ring