Non-Monic polynomials with the same discriminant as the cyclotomic polynomial.

Question

Given a cyclotomic field $K_n=Q(ξ)$ where $ξ$ is a primitive $n$th root of unity, and the class number $h$ of $K_n$ is greater than $1$, what are the non-monic irreducible polynomials $P(x)$ with the same discriminant and degree as $Φ_n(x)$, the $n$th cyclotomic polynomial? For instance take $ξ$, a primitive $39$th root of unity, and $K_{39}=Q(ξ)$, the 39th cyclotomic field. $K$ has class number $h=2$.

$$Φ_{39}(x) = x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} \\ + \: x^{12} - x^{10} + x^9 - x^7 + x^6 - x^4 + x^3 - x + 1$$

it has discriminant $d = 3^{12}*13^{22}$.

Generating other monic irreducible polynomials with this same discriminant are easy, but what about non-monic irreducible polynomials?

$$P(x) = 13*x^{24} + 52*x^{23} - 182*x^{22} - 754*x^{21} + 520*x^{20} \\ + 3172*x^{19} + 3679*x^{18} + 260*x^{17} - 8359*x^{16} - 11895*x^{15} \\+ 6292*x^{14} + 11739*x^{13} + 4863*x^{12} + 2139*x^{11} - 12652*x^{10} \\- 1789*x^9 + 5521*x^8 + 1636*x^7 + 5173*x^6 - 2256*x^5 + 1264*x^4\\ - 564*x^3 + 576*x^2 + 54*x + 79$$

also defines the same field as $Φ_{39}(x)$. Its discriminant, however is not the same as $Φ_{39}(x)$. So now, the question is, what is the smallest irreducible polynomial $C(x)$ of degree 24, (in terms of its coefficients) with discriminant $d = 3^{12}*13^{22}$ and leading coefficient $13$?

Thanks for help.

Answer 1

Not sure how much this helps, but you can do this (reference: Wikipedia): for a cyclotomic polynomial:

• Take a change of the variable $x\to x+\alpha$ where $\alpha\in\mathbb Z$. This will transform the polynomial to another monic polynomial with the same discriminant and the roots that are "shifted" by $\alpha$ so the splitting field stays the same. The constant term is not going to be $\pm 1$ for most of those values of $\alpha$.
• Take a change of the variable $x\to\frac{1}{x}$. This will transform the polynomial to another polynomial with swapped highest term with the constant term, and it most likely won't be monic.

In your example, you can take e.g. $\alpha=2$:

$$p(x)=x^{24}\Phi_{39}\left(\frac{1}{x}+2\right)=x^{24}\left(\frac{1}{x^{24}}+\cdots+\Phi_{39}(2)\right)=1+\cdots+\Phi_{39}(2)x^{24}$$

and, according to the above article, the discriminants of $p$ and $\Phi_{39}$ should be the same.