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Is there an algorithm which "transforms" a polynomial $p(x)$ into $p(x^n)$, using the coefficients of $p$ alone? How is this "operation" called anyway? Kind of like Taylor shift which transforms $p(x)$ into $p(x+a)$.

For example:

$$p(x) = x^2 - 13x + 36$$ $$q(x) = p(x^2) = x^4 - 13x^2 + 36$$

And:

$$p(4) = p(2^2) = q(2) = 0$$ $$p(9) = p(3^2) = q(3) = 0$$

Edit: as per the comment of @SimplyBeautifulArt, it is easy when $n > 1$. But what to do if $0 < n < 1$? For example, how to transform $p(x)$ into $p(\sqrt{x}))$? Is it always possible to do such transformation?

$$p(x) = x^4 - 13x^2 + 36$$ $$q(x) = p(x^{1/2}) = x^2 - 13x + 36$$

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    Well, clearly the coefficients don't really change. – 2017-02-17
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    Do you need $q(x)$ to be a polynomial?If yes then the polynomial $p(x)$ musn't have no odd powers.(for the second example). – 2017-02-17
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    @kingW3 Yes, polynomial. So basically, I just divide the powers' values by $n$, correct? And if such (exact) division wouldn't be possible, the root cannot be taken, yes? – 2017-02-17
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    Yes,if there exist $q$ such that $p(x)=q(x^2)$ then you can take $p(x^{1/2})=q(x)$ and in general if there exists $q$ such that $p(x)=q(x^n)$ then $p(x^{1/n})=q(x)$ – 2017-02-17

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