Let $a$ be a rational number such that the polynomial $P_a=X^3-X-a$ is irreducible, let $\alpha_{a}$ denote a root of $P_a$ and let ${\mathbb K}_a={\mathbb Q}(\alpha_{a})$. Similarly, let $b$ be a rational number such that the polynomial $Q_b=X^3-b$ is irreducible, let $\beta_{b}$ denote a root of $Q_b$ and let ${\mathbb L}_b={\mathbb Q}(\beta_{b})$.
Question 1 : let $f(a)$ be equal to $1$ if ${\mathbb K}_a$ is isomorphic to ${\mathbb L}_b$ for some $b$, and $0$ otherwise. Is $f$ computable?
Question 2 : If ${\mathbb K}_a$ is isomorphic to ${\mathbb L}_b$ , do we have bounds for $|b|$ in terms of $a$ ?
Also, let $\mathbb M$ be any fixed number field of degree $3$ over $\mathbb Q$. Let $g(a)$ be equal to $1$ if ${\mathbb K}_a$ is isomorphic to ${\mathbb M}$, and $0$ otherwise. Question 3 : Is $g$ computable?