For any constant $m$, define,

$$f_n = \frac{x(x^n - 1)}{x - 1}m^{p - n}$$

then for $p=5,7,13$, respectively, one forms the ***p*** th degree polynomial in ***x***,

$$f_5 + 5(6f_4 + 63 f_3 + 260 f_2 + 315 f_1) - 5^3m^6 = 0\tag{1}$$

$$f_7 + 7 (4 f_6 + 46 f_5 + 272 f_4 + 845 f_3 + 1232 f_2 + 574 f_1) - 7^2m^8 = 0\tag{2}$$

$$f_{13}  + 13 (2 f_{12} + 25 f_{11} + 196 f_{10} + 1064 f_9 + 4180 f_8 + 12086 f_7 + 25660 f_6 + 39182 f_5 + 41140 f_4 + 27272 f_3 + 9604 f_2 + 1165 f_1) - 13m^{14} = 0\tag{3} $$

For example, let $m=1$, then the following irreducible eqns *are solvable in radicals*:

$$-125 + 3221 x + 1646 x^2 + 346 x^3 + 31 x^4 + x^5 = 0$$

$$-49 + 20812 x + 16794 x^2 + 8170 x^3 + 2255 x^4 + 351 x^5 + 29 x^6 + x^7=0$$

$$-13 + 2100489 x + 2085344 x^2 + 1960492 x^3 + 1605956 x^4 + 1071136 x^5 + 
  561770 x^6 + 228190 x^7 + 71072 x^8 + 16732 x^9 + 2900 x^{10} + 
  352 x^{11} + 27 x^{12} + x^{13}=0$$

***Questions***: 

1. ***Are these solvable in radicals for ANY $m$***? After testing various $m$, I'm 99.99% sure they are, but it would be nice to know their Galois group. There is an online [Magma calculator][1], but I don't know what command to use. :(

2. Given the Dedekind eta function $\eta(\tau)$, these were found by determining integer relations between $m = \left(\frac{\eta\,(p\tau)}{\eta\,(\tau)}\right)^k$ and $x = \left(\frac{\sqrt{p}\,\eta\,(p^2\tau)}{\eta\,(\tau)}\right)^k$ where $k=\frac{24}{p-1}$. Does the relation $F(m,x)$, as a polynomial in $x$, always have a solvable Galois group for *any prime p*?


  [1]: http://magma.maths.usyd.edu.au/calc/