Is the ring $\mathbb{Q}$ isomorphic to $\operatorname{Frac}(\mathbb{Q}[x])$?

Question

Is the field of rational numbers $\mathbb{Q}$ isomorphic to the fraction field $\operatorname{Frac}(\mathbb{Q}[x])$?

Both are fields, I can't disprove it by some algebraic properties that hold for one but not for another.

• I know that $\operatorname{Frac}(\mathbb{Z}[x]) \cong \operatorname{Frac}(\mathbb{Q}[x])$ so I have tried to find out whether $\operatorname{Frac}(\mathbb{Z}[x]) \cong \mathbb{Q}$, but still have no ideas;

• I have also tried to define $\phi : \mathbb{Q} \to \operatorname{Frac}(\mathbb{Q}[x])$ by $$\phi(p/q)=(x^p-1)/(x^q-1)$$ but it is not a ring homomorphism.

I guess that if $\phi$ is a ring homomorphism, then any $\,p/q\,$ should be mapped to some form of fraction of polynomials $\,p(x)/q(x)\,$ such that the product and sum of two polynomials of those form should have the same form, but I don't know how to do.

Answer 1

No, they are not isomorphic.

There is only one possible morphism $f : \mathbb{Q} \to \operatorname{Frac}(\mathbb{Q}[x]) = \mathbb{Q}(x)$: indeed, since it is a ring morphism it must send $1$ to $1$. Then for all $n \in \mathbb{N}$, $f(n) = f(1+\dots+1) = f(1)+\dots+f(1) = n \cdot f(1) = n$, and so $f(-n) = -f(n) = -n$. Finally, if $p/q$ is some fraction, then $q f(p/q) = f(q \cdot p/q) = f(p) = p \implies f(p/q) = p/q$. So in conclusion, $f$ is necessarily the map that sends a rational $p/q$ to the constant fraction $p/q \in \mathbb{Q}(x)$.

But this $f$ is not an isomorphism (it's not surjective), and since it's the only morphism $\mathbb{Q} \to \operatorname{Frac}(\mathbb{Q}[x])$, the two rings are not isomorphic.

Answer 2

$\mathbb{Q}$ is a prime field and $\mathbb{Q}(x)$ - no, since $\mathbb{Q}\varsubsetneq \mathbb{Q}(x)$. Hence they are non-isomorphic.