What does "A mod P generates the residue class field extension" mean?

Question

We have K and finite algebraic extension L.

P is a prime ideal in $O_{L}$ over prime $p\in O_{K}$ and $A\in O_{L}$.

Then the problem says $\bar{A}:=A ~mod ~P$ generates the residue class field extension $(O_{L}/P)/(O_{K}/p)$. Does that mean the power basis of $\bar{A}$?

Thanks

Answer 1

If $l/k$ is any field extension we say $l$ is generated over $k$ by $\alpha$ if $l=k(\alpha)$. So "generated by" is a field notion here. You're probably thinking of the linear algebra notion of "spanned by." Indeed, the claim $l/k$ is generated by $\alpha$ is equivalent to the powers $\{1,\alpha,\alpha^2,\cdots\}$ spanning $l$ over $k$.