Does this trigonometric pattern continue for all primes $p=6m+1$?

Question

(Revised from its original form.) Consider primes $p=6m+1$, and $p>13$:

$p=19=3\times\color{blue}{6}+1$:

Let $\beta = 2\pi/19.\,$ A root of $x^3+x^2-\color{blue}{6}x-7=0$ is $$x=2\big(\cos(2\beta)+\cos(3\beta)+\cos(5\beta)\big)$$ Note that $2+3=5$, or the last multiplier is the sum of the first two.

$p=31=3\times\color{blue}{10}+1$

Let $\beta = 2\pi/31.\,$ A root of $x^3+x^2-\color{blue}{10}x-8=0$ is $$x=2\big(\cos(2\beta)+\cos(4\beta)+\cos(8\beta)+\cos(16\beta)+\cos(30\beta)\big)$$ Similarly, $2+4+8+16=30$. Note that $\cos(30\beta) =\cos(32\beta)$.

$p=37=3\times\color{blue}{12}+1$

Let $\beta = 2\pi/37.\,$ A root of $x^3+x^2-\color{blue}{12}x+11=0$ is $$x=2\big(\cos(2\beta)+\cos(9\beta)+\cos(12\beta)+\cos(15\beta)+\cos(16\beta)+\cos(54\beta)\big)$$ Again, $2+9+12+15+16=54$.

$p=43=3\times\color{blue}{14}+1$

Let $\beta = 2\pi/43.\,$ A root of $x^3+x^2-\color{blue}{14}x+8=0$ is $$x=2\sum_{k=1}^7\cos\big(2^k\beta)$$ But alternatively, $$x=2\big(\cos(\beta)+\cos(4\beta)+\cos(11\beta)+\cos(16\beta)+\cos(21\beta)+\cos(35\beta)+\cos(88\beta)\big)$$ and, $1+4+11+16+21+35=88$.

Question: Is it true that argument multipliers of the cubic roots obey $\displaystyle\sum_{k=1}^{m-1}a_k=a_m$ for any $p=6m+1$?

Answer 1

made a loop to perform the method of Gauss; this is the short version, primes $p \equiv 1 \pmod 3$ up to $1000$

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jagy@phobeusjunior:~$ ./cubic_cyclic_gauss_loop | grep exps
  x^3 + x^2   - 2 x - 1   p  7 p.root  3 exps 6^k  d = 7^2
  x^3 + x^2   - 4 x + 1   p  13 p.root  2 exps 5^k  d = 13^2
  x^3 + x^2   - 6 x - 7   p  19 p.root  2 exps 8^k  d = 19^2
  x^3 + x^2   - 10 x - 8   p  31 p.root  3 exps 15^k  d = 2^2 * 31^2
  x^3 + x^2   - 12 x + 11   p  37 p.root  2 exps 8^k  d = 37^2
  x^3 + x^2   - 14 x + 8   p  43 p.root  3 exps 2^k  d = 2^2 * 43^2
  x^3 + x^2   - 20 x - 9   p  61 p.root  2 exps 8^k  d = 3^2 * 61^2
  x^3 + x^2   - 22 x + 5   p  67 p.root  2 exps 3^k  d = 3^2 * 67^2
  x^3 + x^2   - 24 x - 27   p  73 p.root  5 exps 7^k  d = 3^2 * 73^2
  x^3 + x^2   - 26 x + 41   p  79 p.root  3 exps 12^k  d = 79^2
  x^3 + x^2   - 32 x - 79   p  97 p.root  5 exps 19^k  d = 97^2
  x^3 + x^2   - 34 x - 61   p  103 p.root  5 exps 3^k  d = 3^2 * 103^2
  x^3 + x^2   - 36 x - 4   p  109 p.root  6 exps 2^k  d = 2^4 * 109^2
  x^3 + x^2   - 42 x + 80   p  127 p.root  3 exps 5^k  d = 2^2 * 127^2
  x^3 + x^2   - 46 x + 103   p  139 p.root  2 exps 8^k  d = 139^2
  x^3 + x^2   - 50 x - 123   p  151 p.root  6 exps 3^k  d = 3^2 * 151^2
  x^3 + x^2   - 52 x + 64   p  157 p.root  5 exps 2^k  d = 2^4 * 157^2
  x^3 + x^2   - 54 x - 169   p  163 p.root  2 exps 5^k  d = 163^2
  x^3 + x^2   - 60 x - 67   p  181 p.root  2 exps 6^k  d = 5^2 * 181^2
  x^3 + x^2   - 64 x + 143   p  193 p.root  5 exps 11^k  d = 3^2 * 193^2
  x^3 + x^2   - 66 x + 59   p  199 p.root  3 exps 12^k  d = 5^2 * 199^2
  x^3 + x^2   - 70 x - 125   p  211 p.root  2 exps 8^k  d = 5^2 * 211^2
  x^3 + x^2   - 74 x - 256   p  223 p.root  3 exps 13^k  d = 2^2 * 223^2
  x^3 + x^2   - 76 x - 212   p  229 p.root  6 exps 2^k  d = 2^4 * 229^2
  x^3 + x^2   - 80 x + 125   p  241 p.root  7 exps 17^k  d = 5^2 * 241^2
  x^3 + x^2   - 90 x + 261   p  271 p.root  6 exps 24^k  d = 3^2 * 271^2
  x^3 + x^2   - 92 x + 236   p  277 p.root  5 exps 2^k  d = 2^4 * 277^2
  x^3 + x^2   - 94 x + 304   p  283 p.root  3 exps 2^k  d = 2^2 * 283^2
  x^3 + x^2   - 102 x - 216   p  307 p.root  5 exps 2^k  d = 2^2 * 3^2 * 307^2
  x^3 + x^2   - 104 x + 371   p  313 p.root  10 exps 7^k  d = 313^2
  x^3 + x^2   - 110 x - 49   p  331 p.root  3 exps 7^k  d = 7^2 * 331^2
  x^3 + x^2   - 112 x + 25   p  337 p.root  10 exps 5^k  d = 7^2 * 337^2
  x^3 + x^2   - 116 x - 517   p  349 p.root  2 exps 6^k  d = 349^2
  x^3 + x^2   - 122 x + 435   p  367 p.root  6 exps 3^k  d = 3^2 * 367^2
  x^3 + x^2   - 124 x - 221   p  373 p.root  2 exps 8^k  d = 7^2 * 373^2
  x^3 + x^2   - 126 x + 365   p  379 p.root  2 exps 8^k  d = 5^2 * 379^2
  x^3 + x^2   - 132 x - 544   p  397 p.root  5 exps 15^k  d = 2^4 * 397^2
  x^3 + x^2   - 136 x - 515   p  409 p.root  21 exps 11^k  d = 5^2 * 409^2
  x^3 + x^2   - 140 x - 343   p  421 p.root  2 exps 8^k  d = 7^2 * 421^2
  x^3 + x^2   - 144 x - 16   p  433 p.root  5 exps 42^k  d = 2^6 * 433^2
  x^3 + x^2   - 146 x - 504   p  439 p.root  15 exps 3^k  d = 2^2 * 3^2 * 439^2
  x^3 + x^2   - 152 x - 220   p  457 p.root  13 exps 5^k  d = 2^6 * 457^2
  x^3 + x^2   - 154 x + 343   p  463 p.root  3 exps 7^k  d = 7^2 * 463^2
  x^3 + x^2   - 162 x - 505   p  487 p.root  3 exps 7^k  d = 7^2 * 487^2
  x^3 + x^2   - 166 x + 536   p  499 p.root  7 exps 2^k  d = 2^2 * 3^2 * 499^2
  x^3 + x^2   - 174 x - 891   p  523 p.root  2 exps 8^k  d = 3^2 * 523^2
  x^3 + x^2   - 180 x + 521   p  541 p.root  2 exps 8^k  d = 7^2 * 541^2
  x^3 + x^2   - 182 x - 81   p  547 p.root  2 exps 8^k  d = 3^4 * 547^2
  x^3 + x^2   - 190 x - 719   p  571 p.root  3 exps 7^k  d = 7^2 * 571^2
  x^3 + x^2   - 192 x + 171   p  577 p.root  5 exps 14^k  d = 3^4 * 577^2
  x^3 + x^2   - 200 x + 512   p  601 p.root  7 exps 31^k  d = 2^6 * 601^2
  x^3 + x^2   - 202 x - 1169   p  607 p.root  3 exps 6^k  d = 607^2
  x^3 + x^2   - 204 x + 999   p  613 p.root  2 exps 8^k  d = 3^2 * 613^2
  x^3 + x^2   - 206 x + 321   p  619 p.root  2 exps 3^k  d = 3^4 * 619^2
  x^3 + x^2   - 210 x - 1075   p  631 p.root  3 exps 27^k  d = 5^2 * 631^2
  x^3 + x^2   - 214 x - 1024   p  643 p.root  11 exps 2^k  d = 2^2 * 3^2 * 643^2
  x^3 + x^2   - 220 x - 1273   p  661 p.root  2 exps 8^k  d = 3^2 * 661^2
  x^3 + x^2   - 224 x - 997   p  673 p.root  5 exps 10^k  d = 7^2 * 673^2
  x^3 + x^2   - 230 x + 128   p  691 p.root  3 exps 2^k  d = 2^2 * 5^2 * 691^2
  x^3 + x^2   - 236 x + 1313   p  709 p.root  2 exps 8^k  d = 709^2
  x^3 + x^2   - 242 x + 1104   p  727 p.root  5 exps 3^k  d = 2^2 * 3^2 * 727^2
  x^3 + x^2   - 244 x + 1276   p  733 p.root  6 exps 2^k  d = 2^4 * 733^2
  x^3 + x^2   - 246 x - 520   p  739 p.root  3 exps 2^k  d = 2^2 * 5^2 * 739^2
  x^3 + x^2   - 250 x + 1057   p  751 p.root  3 exps 6^k  d = 7^2 * 751^2
  x^3 + x^2   - 252 x + 729   p  757 p.root  2 exps 8^k  d = 3^4 * 757^2
  x^3 + x^2   - 256 x - 1481   p  769 p.root  11 exps 7^k  d = 5^2 * 769^2
  x^3 + x^2   - 262 x - 991   p  787 p.root  2 exps 3^k  d = 3^4 * 787^2
  x^3 + x^2   - 270 x + 1592   p  811 p.root  3 exps 2^k  d = 2^2 * 811^2
  x^3 + x^2   - 274 x + 61   p  823 p.root  3 exps 5^k  d = 11^2 * 823^2
  x^3 + x^2   - 276 x - 307   p  829 p.root  2 exps 7^k  d = 11^2 * 829^2
  x^3 + x^2   - 284 x + 1011   p  853 p.root  2 exps 5^k  d = 3^4 * 853^2
  x^3 + x^2   - 286 x - 509   p  859 p.root  2 exps 8^k  d = 11^2 * 859^2
  x^3 + x^2   - 292 x + 1819   p  877 p.root  2 exps 6^k  d = 877^2
  x^3 + x^2   - 294 x + 1439   p  883 p.root  2 exps 8^k  d = 7^2 * 883^2
  x^3 + x^2   - 302 x - 739   p  907 p.root  2 exps 8^k  d = 11^2 * 907^2
  x^3 + x^2   - 306 x - 1872   p  919 p.root  7 exps 3^k  d = 2^2 * 3^2 * 919^2
  x^3 + x^2   - 312 x - 2221   p  937 p.root  5 exps 15^k  d = 937^2
  x^3 + x^2   - 322 x + 1361   p  967 p.root  5 exps 3^k  d = 3^4 * 967^2
  x^3 + x^2   - 330 x - 2349   p  991 p.root  6 exps 3^k  d = 3^2 * 991^2
  x^3 + x^2   - 332 x - 480   p  997 p.root  7 exps 2^k  d = 2^4 * 3^2 * 997^2
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Answer 2

This situation can be summarized as follows (I take the case p = 19 as an example): Let $\zeta$ be a $19$-th root of unity then we can form the field of cyclotomic numbers $\mathbb{Q}(\zeta)$. The galois group of this extension (of $\mathbb{Q}$) is the cyclic group $C_{18}$ of order $18$. In order to construct a sub-extension $K$ of dimension $3$, lets chose a subgroup $H$ of $G$ of index $3$, i.e. the cyclic subgroup of order $18/3 = 6$. The orbit of $\zeta^2$ gives $\{\zeta^2, \zeta^3, \zeta^5, \ldots \text{(complex conjugates)}\}$. The sum $x$ of these elements is invariant under $H$ so belongs to $K$, so it's minimal polynomial is cubic : $x^3 + x^2 -6x-7$.The same situation reproduces for the other values of $p$, so the problem rather concerns the powers of a generator in orbits of cosets of subgroups in the automorphism group of the cyclic group $C_p$.