Let $ F $ be a field and $ K $ be an extension of $ F $. Define the set, $$ E(n, K, F) := \{ M \in GL(n, K) , \det M \in F \} $$
Show that $ E(n, K, F) $ is a normal subgroup of $ G(n, K) $ and also determine the quotient $ G(n, K) / E(n, K, F) $.
My idea : Showing the normality is easy. Also, I can prove that $$ G(n, K) / E(n, K, F) \simeq K^{ \times }/ F^{ \times } $$ by using the determinant map $ \psi : GL(n, K) \rightarrow K^{ \times} , \psi(M) = \det M, M \in GL(n, K) $, noticing that $ \psi $ maps $ E(n, K, F) $ onto $ F^{ \times } $ and using the isomorphism theorem for normal subgroups.
However, I still do not get much insight about the structure. Is there any place else where $ E(n, K, F) $ appears?