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

[TopCat]

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?

 

[morphism]

Hungerford's x_i's are supposed to be indeterminates, i.e. they're transcendental over K by definition (essentially).

When you write down "i" you're implicitly thinking of a complex number that satisfies the polynomial x^2+1 - i.e., you're not thinking of an indeterminate.