field of fractions and being algebraically closed

Question

prove that for every field $F$ the field of fractions $F(x)$ is not algebraically closed.

it is a problem which i don't know how to deal with it.

help please.

thank you.

Answer 1

By definition

$$F(x)=\left\{\;\;\frac{f(x)}{g(x)}\;;\;\;f,g\in F[x]\;,\;\;g(x)\neq 0\right\}$$

If we had

$$\left(\frac{f(x)}{g(x)}\right)^2=x\iff f(x)^2=xg(x)^2$$

But $\;\deg f(x)^2\;$ is even, whereas $\;\deg (xg(x))\;$ is odd and you're done.

Answer 2

Consider the polynomial $S(x,y)=y^2-x$ this does not have a root in $K(x)$. By irreducibility criteria and Gauss Lemma.