Transcendental extension over a field K.

Question

Prove that $x$ is transcendental over $F(x)$ or more generally show that any element not in $K$ but in $K(x_1,x_2,x_3,x_4,\ldots,x_n)$ is transcendental?

Answer 1

Suppose the reduced fraction ( reduced in the ring $\;K[x_1,..,x_n]\;$ )

$$\frac pq=\frac{p(x_1,..,x_n)}{q(x_1,...,x_n)}\in K(x_1,...,x_n)\;\;\text{is algebraic over $\;K\;$ , say}\;$$

$$\sum_{i=0}^m k_i\left(\frac pq\right)^i=0\implies k_0q^m+k_1pq^{m-1}+\ldots+k_mp^m=0$$

since $\forall\,1\le i\le m\;,\; p\mid k_ip^{i}q^{m-i}\;$ , we get that it must be $\;p\mid k_0q^m\;\implies p\;$ is a constant.

Now prove in a similar form that also $\;q\;$ must be a constant, from which we get that $\;\frac pq\in K\;$ and we're done.