The characteristic of a subfield is the same as the characteristic of the larger field.
Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
A simple extension is algebraic if and only if it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic.See transcendence degree for examples and more extensive discussion of transcendental extensions.
A consequence of the primitive element theorem states that every finite separable extension has a primitive element (i.e. is simple).