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.


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