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In the field extension

[tex]K \subset K(x_{1},...,x_{n})[/tex]

each [itex]x_{i}[/itex] is easily seen to be transcendental over K. In fact, every element of [itex]K(x_{1},...,x_{n})[/itex] not in K itself is transcendental over K.

But if we take K = ℝ and [itex]K(x_{1})[/itex] = 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?