I guess that $Q$ and $Q(\pi)$ are not isomorphic as fields. And I know $Q(\pi)$ is isomorphic to $Q(x)$. So it seems to suffice to show that $Q(x)$ is not isomorphic to $Q$ as fields.
But I really don’t know how to show two fields are not isomorphic.
My trial was to find a polynomial that has roots in $Q(x)$, but not in $Q$. Sadly, I couldn’t come up with anything.
Can somebody help me? Thank you.
As Thomas Andrews notes in the comments, $\mathbb Q$ is a nontrivial subfield of $\mathbb Q(\pi)$, but $\mathbb Q$ has no nontrivial subfields. However, there do exist fields $K$ for which $K$ and $K(x)$ are isomorphic: consider $$K=\mathbb Q(x_1,x_2,\dots)$$ consisting of all rational functions with coefficients in $\mathbb Q$ and countably many variables $x_i$ for each $i\in\mathbb N$. There is an isomorphism between $K$ and $$K(x)=\mathbb Q(x,x_1,x_2,\dots)$$ given by $x_1\to x$, $x_2\to x_1$, $x_3\to x_2$, etc.