# $F_{\sigma}$ is in the group of automorphisms of $V$. What does it mean to say that $F_{\sigma}$ is defined over $k$?

In André Weil's paper 'Algebras with involution and the classical groups', while setting up basic result for cocyles, he defines what a cocyle is, and the definition is as follows:

By a cocycle, we shall understand a mapping $$\sigma \rightarrow F_{\sigma}$$ of $$\mathfrak{g}$$ into the group of automorphisms of the vector-space $$V$$, such that: (a) for each $$\sigma \in \mathfrak{g}$$, $$F_{\sigma}$$ is defined over $$K$$; (b) for all $$\sigma, \tau \in \mathfrak{g}$$ we have $$F_{\sigma\tau} = (F_{\sigma})^{\tau} \circ F_{\tau}$$.

Here is the background setting: $$U$$ is a fixed universal domain of characteristic zero. $$k$$ is a fixed groundfield inside $$U$$, and $$K$$ is a normal extension of $$k$$ of finite degree $$d$$. $$\mathfrak{g}$$ denotes the Galois group of $$K$$ over $$k$$. $$V$$ is a vector space of dimension $$n$$ over the universal domain.

I have trouble understanding (a) part of this definition. $$F_{\sigma} : V \rightarrow V$$ is an automorphism, so I assume it is $$U$$-linear. So, what does it mean for $$F_{\sigma}$$ to be defined over $$K$$? It is not just $$K$$-linear I think because $$F_{\sigma}$$ will always be $$U$$-linear hence $$K$$-linear.