Definition
Let be a field extension.
An element is called algebraic (over ) if the degree of the simple extension and transcendential otherwise.
The extension is called algebraic if all of its elements are algebraic.
Relation to roots
For an extension , is algebraic if and only if is a root of a non-zero polynomial with coefficients.
This follows from this theorem.
This theorem aligns with the general notion of algebraic numbers, though the given definition is a generalization
Finite extensions
Finite extensions are algebraic.
Proof
Let be a finite extension.
If , then is a -vector subspace.
therefore, is algebraic for all .
Transitivity of algebraic
Let be extensions.
Then is algebraic if and only if and are algebraic.
Proof
Assume is algebraic, then each is a root of some .
Therefore, is algebraic as .
Also, is algebraic as for every , .
Next assume and are algebraic.
Let .
The assumption that is algebraic means there is some polynomial
such that is a root.
is a polynomial with coefficients in , but over these coefficients might not exist.
Instead, we can add these coefficients into then
where is finitely generated.
Now, is algebraic over .
Since is algebraic, then is algebraic (by logic from first part of proof), and thus is finite.
Hence, are algebraic over .
This means that
is finite since is a root of so is finite so is algebraic over .
References
@aluffi2009Algebra - Chapter VII.2