Relation between general linear groups for subfield $K$ of $F$ of finite index.

Question

If $F$ is a field and $d \ge 1$.

Let $K$ be a subfield of $F$ with finite index $k = [F : K]$. Then $F$ is a $k$-dimensional vector space over $K$. Thus every $F$-vector space is also a $K$-vector space and any $F$-linear transformation is also $K$-linear. Specifically this means that $F^d$ is isomorphic to $K^{kd}$ as a $K$-vector space and that $GL_d(F)$ is isomorphic to a subgroup of $GL_{kd}(K)$.

This is taken from page 55 of Dixon & Mortimer Permutation Groups. I ask myself why $GL_d(F)$ is just isomorphic to a subgroup of $GL_{kd}(K)$, can you give an example of an element from $GL_{kd}(K)$ not contained in $GL_d(F)$ under the identification $F^d$ with $K^{kd}$?

Answer 1

Every $F$-linear automorphism will be $K$-linear, but the converse need not be true. For instance, if you consider $\mathbb C$ as a $\mathbb C$-vector space, then it is one-dimensional. The only $\mathbb C$-linear automorphisms are therefore given by multiplication by a nonzero complex scalar. However, if you consider $\mathbb C$ as $\mathbb R$-vector space, there exist other $\mathbb R$-linear automorphisms. For instance, complex conjugation is one such automorphism.

In more generality, any nontrivial element of $\operatorname{Gal}(F/K)$ by definiton gives a $K$-linear automorphism of $F$ which is not $F$-linear. You can build higher dimensional examples from these.