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This is a kind of meta-question, influenced by this and the desire for recognition of a general rather than a specific answer - i.e. that it can always be done, and there is a method which always works, even though in specific cases there may be tricks to shorten the solution.

Suppose we have an irreducible polynomial $p(x)$ which might typically be monic with integer coefficients and degree $n$.

Say that a typical root is $a$ and we want to find a polynomial $q(y)$ which has as its roots $r(a)$ where $r$ is some rational function like $a^2+a$ or $\cfrac 1{a^2+1}$.

Since $q$ will in general have $n$ roots, we see that unless $r$ has special features in relation to $p$, we expect a polynomial of degree $n$.

Two questions, therefore:

1 What can we say about the coefficients of $q$ - are there conditions for these to be integers, and in what circumstances might $q$ also be monic?

2 How can we best find $q$? - What practical general methods exist which would enable a student to compute $q$? - the linked question has two approaches, one via symmetric polynomials and functions and the other in relation to linear maps and determinants. It would be good to see these well explained, and any other ideas too.

I am particularly looking for answers which will help with actual computations, and/or which may help motivate students to understand and investigate general theorems about field extensions and the like and put particular problems they have been set into a richer mathematical context.

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    $\begingroup$ One could get a polynomial equation in $y = \frac{a(x)}{b(x)}$ by using resultants to eliminate $x$ between: $$ \begin{cases} p(x) = 0 \\ b(x) \cdot y -a(x) = 0 \end{cases} $$ $\endgroup$ – dxiv Feb 4 '17 at 23:04
  • $\begingroup$ A classical keyword for this is "transformation". See for example (sites.google.com/site/mathcuts/…). $\endgroup$ – Jean Marie Feb 4 '17 at 23:11
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If $p$ is monic with integer coefficients, then $a$ is an algebraic integer.

If $r$ is a polynomial, then $r(a)$ is an algebraic integer and so $q$ can be chosen to be monic with integer coefficients.

If $r$ is a rational function, then $r(a)$ is an algebraic number and $q$ can be chosen to have integer coefficients, but it won't be monic in general.

The method using the matrix of the map $x \mapsto ax$ described in my answer is systematic and general. It leverages knowledge of linear algebra, which presumably came first.

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