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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?

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    Some of these results such as the isomorphisms may already be known in algebra. First, since these are countable field adjoining a single element, you can come up with a reasonable definition of function between them being computable. Then look at the classical algebra proof. Usually, these isomorphism are defined by mapping some element to another element and extending linearly. Such a map will probably be computable.2011-12-01
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    @ William : you were misled by the original title (I changed it), my question is not about computing the isomorphism, but about deciding if an isomorphism exists or not.2011-12-01

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