Is $\mathbb{Q}[1]$ isomorphic to $\mathbb{Q}$?

Question

If I am not mistaken, $\mathbb{Q}[1] \cong \mathbb{Q}/(f(X))$ denotes the ring of polynomials with rational coefficients evaluated in $1$ ($1$ being the root of the polynomial $f(X)$ as well). Could we say that there exists an isomorphism such that $\mathbb{Q}[1] \cong \mathbb{Q}$? It seems to me that it's not impossible to express any fraction as a sum of fractions and vice-versa. If such an isomorphism exists, what does it look like? And if not, why doesn't it exist?