An algebraic extension 
of a field 
has every element of 
algebraic over 
(i.e., each element is the root of a nonzero polynomial with
coefficients in 
).
An algebraic extension need not be a finite extension. It is finite iff it is finitely generated as a field
over 
. Indeed, adjoining finitely many algebraic elements
gives a finite tower of finite extensions, whereas
any finite extension is generated as a field
by a vector basis (Milne 2022, p. 19).
Algebraicity is transitive: if 
and both 
and 
are algebraic, then 
is algebraic (Milne 2022, pp. 19-20).
