I don't think that's the usual definition of algebric extension. Wikipedia (https://en.wikipedia.org/wiki/Algebraic_extension https://en.wikipedia.org/wiki/Alge

I don't think that's the usual definition of algebric extension. Wikipedia (https://en.wikipedia.org/wiki/Algebraic_extension https://en.wikipedia.org/wiki/Algebraic_extension) says that an algebraic extension is one where every element is the root of some nonzero polynomial over the base field. So for example the algebraic numbers would be algebraic over the rationals, even though they're infinite dimensional over the rationals.