The relation between $GF(2)$ and $GF(2^3)$

Question

Both $GF(2)$ and $GF(2^3)$ are finite fields of characteristic $2$.

Is $GL(2^3)$ an extension of $GF(2)$?

Can someone point some links that details something about this, please?

Answer 1

This can be found in nearly any discussion of finite fields.

Here's a standard way to realize an embedding $GL(2) \hookrightarrow GL(2^3)$. Since $p(x):=x^3+x+1$ is irreducible in $GF(2)$, the ideal it generates in $GF(2)[x]$ is maximal. Thus, $GF(2)[x]/\langle p(x)\rangle$ is a field with $2^3$ elements and so we may identify it with $GF(2^3)$. In particular, it contains $GF(2)$ as the subfield of (equivalence classes of) constant polynomials.

In general one can play the same game to realize $GL(p^{kn})$ as an extension of $GL(p^n)$ for any $p, n, k$.

For a little more about this, see, e.g., Dummit & Foote, $\S$13.