# Finding a $\gamma$ to define a number field like $E=\mathbb{Q}(\zeta_5)(X^5-\gamma)$

By working with eliptic curves, I found that the extension E defined by:

E.<a> = NumberField(x^20 - 2*x^19 - 2*x^18 + 18*x^17 - 32*x^16 + 88*x^15 + 58*x^14 -
782*x^13 + 1538*x^12 + 1348*x^11 - 466*x^10 - 894*x^9 + 346*x^8 -
114*x^7 - 424*x^6 - 88*x^5 + 214*x^4 + 54*x^3 + 14*x^2 + 4*x + 1)


Is a cyclic Kummer extension of degree $5$ over $\mathbb{Q}(\zeta_5)$, thus by the clasification of Kummer extensions, there exists $\gamma \in \mathbb{Q}(\zeta_5)$ such that $E=\mathbb{Q}(\sqrt{\gamma})$.

So, about all the polynomials $f$ that define $E$ how do I find $\gamma$ such that the polynomial $X^5-\gamma$ also defines $E$?

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From the discriminant of $f$ you know that $\gamma$ is an S-unit where S contains the primes above 5,11. You can compute the relevant S-unit group. Since $\gamma$ is only determined modulo 5th power, this already gives you a finite list of candidates. The list might be a bit long to loop through, so you might want to use some linear algebra and reduction information to cut down the list a little bit.

You can determine a large list of primes that split completely in your field E quite easily (just loop through the primes and keep the ones where the reduction of your given $f$ has a root). At each of those primes your $\gamma$ needs to reduce to a fifth power. That gives you a linear condition mod 5 on the exponent vector wrt. your generating set for S-units. This quite quickly cuts down the list to find that a $\gamma$ with minimal polynomial

x^2-1661*x-161051


has the property you're looking for (that's to say: x^10-1661*x^5-161051 has a root in E).

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Mmm sorry I did not understand what you told me, shall I take primes from $\mathbb{Q}(\zeta_5)$ and make them split completely in $E$? And I don't know what is the S-unit group.

You should definitely look up the definition of S-units, because they are essential for this type of problem. Any good text on computational algebraic number theory will devote quite some effort on them.

The argument I used is similar to how you would know, if I tell you a quadratic extension is unramified outside of 2,3, you know it must be Q(sqrt(d)) with d in {-1,2,-2,3,-3,6,-6}. Splitting information allows you to further cut down on the candidates.

Constructive Class field theory and Kummer theory are the relevant keyworks to do a literature search on.