# When is an element of a finitely generated field extension algebraic?

Hungerford says that
In the field extension
$$K \subset K(x_{1},...,x_{n})$$
each $x_{i}$ is easily seen to be transcendental over K. In fact, every element of $K(x_{1},...,x_{n})$ not in K itself is transcendental over K.

But if we take K = ℝ and $K(x_{1})$ = ℝ(i) = ℂ, we have that i is not in ℝ yet is algebraic over ℝ. Guess I'm missing something here. Is it that this need not be true for simple extensions if the primitive element is algebraic over the field?