It is possible to adjoin an element $\alpha$ to a field $\mathbb{F}$. Is there a name for the operation to take away an element $\alpha$ (and all subsequent elements) from a field $\mathbb{F}$ to form a subfield $\mathbb{E}$?
I came across this notion when trying to visualize the field $\hat{\mathbb{Z}}_2$, the algebraic closure of $\mathbb{Z}_2$. The closure is formed by consecutively adjoining roots of polynomials in $\mathbb{Z}_2$ or some extensions of $\mathbb{Z}_2$. However, there are infinitely such polynomials, and thus infinitely such roots to adjoin, which makes it hard to visualize. I think it may be interesting trying to take away elements from $\hat{\mathbb{Z}}_2$ instead.