This question is from Rotman's Group Theory. This is more of a "If I knew what that symbol/word/phrase meant I might understand the whole Idea" question and less of a "I am so lost I cannot even understand the question" question.
Assume $F$ is a subfield of $C$ and $\{\alpha_1,...,\alpha_n\}\subset C.$ If $\sigma_i:F(\alpha_1,...,\alpha_n)\to C,i=1,2,$ are field maps with $\sigma_1|F=\sigma_2|F$ and $\forall i \ \sigma_1(\alpha_i)=\sigma_2(\alpha_i)$, then $\sigma_1=\sigma_2$
What does $\sigma_1|F$ mean? My guess is it means either extended image or restricted image.
It means the same map as $\sigma_1$ but now with domain only $F$ rather than the original (bigger) domain $F(\alpha_1,\ldots,\alpha_n)$.
This is usually called the restriction to $F$.