Is F[x] isomorphic to Z for any field F?

Question

I am wondering if the ring of polynomials $F[x]$ with coefficients in the field $F$ is ever isomorphic to $\mathbb{Z}$ for some field $F$.

For all the fields I've examined, such as $F = \mathbb{C}$ or $F = \mathbb{R}$, it's true that both $\mathbb{C}[x]$ and $\mathbb{R}[x]$ cannot be isomorphic to $\mathbb{Z}$. Is it true that $F[x]$ is not isomorphic to $\mathbb{Z}$ for any field $F$ ? If so, is there a nice way to prove this in general?

Thanks!

Answer 1

If there was such an isomorphism of rings then the additive groups would be isomorphic. However, the additive group of $\mathbb{Z}$ is cyclic. Now, can the additive group of $F[x]$ be cyclic?

Answer 2

No such fields exist.

If $\phi: F[x] \to \mathbb Z$ is a ring homomorphism, then so is its restriction to $F$. This restriction must be an injection. Therefore, $\phi(F)$ is a subfield of $\mathbb Z$, which has no subfields.