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Thursday, Novemeber 10, 2016: I found the method of Gauss written up in Galois Theory by David Cox, chapter 9, section 2. The method lends itself to computer programming, otherwise your eyes start to blur after a few of the calculations. I posted an answer with polynomials extending the primes up to 311. Gauss, clever guy.

To save space, here are just the polynomials for primes $p \equiv 1 \pmod {10}$ up to $1000.$

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jagy@phobeusjunior:~$ ./quintic_cyclic_gauss_loop | grep exps
  x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1   p  11 p.root  2 exps 10^k  d  =  11^4
  x^5 + x^4 - 12 x^3 - 21 x^2 + 1 x + 5   p  31 p.root  3 exps 6^k  d  =  5^2 31^4
  x^5 + x^4 - 16 x^3 + 5 x^2 + 21 x - 9   p  41 p.root  6 exps 3^k  d  =  3^6 41^4
  x^5 + x^4 - 24 x^3 - 17 x^2 + 41 x - 13   p  61 p.root  2 exps 21^k  d  =  29^2 61^4
  x^5 + x^4 - 28 x^3 + 37 x^2 + 25 x + 1   p  71 p.root  7 exps 23^k  d  =  23^2 71^4
  x^5 + x^4 - 40 x^3 + 93 x^2 - 21 x - 17   p  101 p.root  2 exps 32^k  d  =  17^2 101^4
  x^5 + x^4 - 52 x^3 - 89 x^2 + 109 x + 193   p  131 p.root  2 exps 18^k  d  =  79^2 131^4
  x^5 + x^4 - 60 x^3 - 12 x^2 + 784 x + 128   p  151 p.root  6 exps 23^k  d  =  2^18 151^4
  x^5 + x^4 - 72 x^3 - 123 x^2 + 223 x - 49   p  181 p.root  2 exps 17^k  d  =  7^2 149^2 181^4
  x^5 + x^4 - 76 x^3 - 359 x^2 - 437 x - 155   p  191 p.root  19 exps 11^k  d  =  5^2 11^2 191^4
  x^5 + x^4 - 84 x^3 - 59 x^2 + 1661 x + 269   p  211 p.root  2 exps 26^k  d  =  31^2 67^2 211^4
  x^5 + x^4 - 96 x^3 - 212 x^2 + 1232 x + 512   p  241 p.root  7 exps 11^k  d  =  2^16 11^2 241^4
  x^5 + x^4 - 100 x^3 - 20 x^2 + 1504 x + 1024   p  251 p.root  6 exps 2^k  d  =  2^18 5^4 251^4
  x^5 + x^4 - 108 x^3 - 401 x^2 - 13 x + 845   p  271 p.root  6 exps 12^k  d  =  5^2 13^4 271^4
  x^5 + x^4 - 112 x^3 - 191 x^2 + 2257 x + 967   p  281 p.root  3 exps 6^k  d  =  193^2 281^4
  x^5 + x^4 - 124 x^3 + 535 x^2 - 413 x - 539   p  311 p.root  17 exps 11^k  d  =  7^4 13^2 311^4
  x^5 + x^4 - 132 x^3 - 887 x^2 - 1843 x - 1027   p  331 p.root  3 exps 13^k  d  =  13^2 31^2 331^4
  x^5 + x^4 - 160 x^3 + 369 x^2 + 879 x - 29   p  401 p.root  3 exps 26^k  d  =  29^2 401^4 433^2
  x^5 + x^4 - 168 x^3 + 219 x^2 + 3853 x - 3517   p  421 p.root  2 exps 32^k  d  =  223^2 239^2 421^4
  x^5 + x^4 - 172 x^3 - 724 x^2 + 1824 x + 1728   p  431 p.root  7 exps 47^k  d  =  2^20 3^4 431^4
  x^5 + x^4 - 184 x^3 - 129 x^2 + 4551 x + 5419   p  461 p.root  2 exps 13^k  d  =  163^2 461^4 491^2
  x^5 + x^4 - 196 x^3 + 59 x^2 + 2019 x + 1377   p  491 p.root  2 exps 32^k  d  =  3^4 17^2 229^2 491^4
  x^5 + x^4 - 208 x^3 - 771 x^2 + 4143 x + 2083   p  521 p.root  3 exps 24^k  d  =  61^2 521^4 577^2
  x^5 + x^4 - 216 x^3 + 1147 x^2 - 805 x - 2629   p  541 p.root  2 exps 11^k  d  =  11^2 311^2 541^4
  x^5 + x^4 - 228 x^3 + 868 x^2 + 3056 x - 7552   p  571 p.root  3 exps 2^k  d  =  2^22 31^2 571^4
  x^5 + x^4 - 240 x^3 + 1755 x^2 - 3731 x + 2399   p  601 p.root  7 exps 17^k  d  =  5^2 13^2 17^2 601^4
  x^5 + x^4 - 252 x^3 + 2095 x^2 - 5785 x + 5069   p  631 p.root  3 exps 24^k  d  =  89^2 631^4
  x^5 + x^4 - 256 x^3 - 564 x^2 + 5328 x - 5120   p  641 p.root  3 exps 21^k  d  =  2^16 5^2 61^2 641^4
  x^5 + x^4 - 264 x^3 - 185 x^2 + 16837 x + 4851   p  661 p.root  2 exps 32^k  d  =  3^16 7^2 661^4
  x^5 + x^4 - 276 x^3 - 1299 x^2 + 5329 x + 15581   p  691 p.root  3 exps 11^k  d  =  379^2 397^2 691^4
  x^5 + x^4 - 280 x^3 + 2047 x^2 - 3791 x + 1699   p  701 p.root  2 exps 23^k  d  =  17^2 19^2 23^2 701^4
  x^5 + x^4 - 300 x^3 - 2313 x^2 - 3761 x - 571   p  751 p.root  3 exps 11^k  d  =  41^2 631^2 751^4
  x^5 + x^4 - 304 x^3 + 2831 x^2 - 8925 x + 8775   p  761 p.root  6 exps 3^k  d  =  3^4 5^2 23^2 761^4
  x^5 + x^4 - 324 x^3 - 3471 x^2 - 12431 x - 13603   p  811 p.root  3 exps 12^k  d  =  7^4 47^2 811^4
  x^5 + x^4 - 328 x^3 - 1215 x^2 + 3573 x + 2179   p  821 p.root  2 exps 32^k  d  =  37^4 109^2 821^4
  x^5 + x^4 - 352 x^3 - 2361 x^2 + 4257 x + 9967   p  881 p.root  3 exps 29^k  d  =  29^2 881^4 953^2
  x^5 + x^4 - 364 x^3 - 2988 x^2 - 1392 x + 9856   p  911 p.root  17 exps 22^k  d  =  2^18 7^2 11^2 911^4
  x^5 + x^4 - 376 x^3 + 3877 x^2 - 13445 x + 15271   p  941 p.root  2 exps 12^k  d  =  191^2 941^4
  x^5 + x^4 - 388 x^3 + 1476 x^2 + 8304 x + 7168   p  971 p.root  6 exps 2^k  d  =  2^20 7^2 13^2 971^4
  x^5 + x^4 - 396 x^3 + 2101 x^2 + 8039 x - 1819   p  991 p.root  6 exps 30^k  d  =  107^2 991^4 1399^2

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ORIGINAL QUESTION: I should emphasize that I am interested in quintics with five real irrational roots, Galois group cyclic $\mathbb Z_5,$ such that the roots can be expressed as sums of roots of unity (in conjugate pairs), therefore as sums of cosines of rational multiples of $\pi.$

The wording at the wikipedia section on solvable cyclic quintics suggests that there is an infinite sequence of such examples as they display. However, they give no source for this sub-section. As you will see below, i had no trouble extending their recipe for primes $101$ and $131,$ but the items i found for prime $151$ have the wrong Galois groups. I have jumped to prime $181$ and bigger $p = 10 n + 1,$ we will see what happens. I have stuck with their recipe...note that, by a strict integer translation, it is reasonable to also consider $x^5 + 2 x^4 + more,$ or $x^5 + 3 x^4 + more,$ or $x^5 + 4 x^4 + more.$ There is plenty of literature on $x^5 + e x^3 + stuff.$ Oh, with $p = 10 n + 1$ prime, our polynomial is $$ x^5 + x^4 - 4 n x^3 + a x^2 + b x + c $$ i have been assuming that we want the discriminant to be a square, in particular $w^2 p^4,$ where $w$ is not divisible by $p.$

The questions are: where did wikipedia find this material, also are there more.

https://en.wikipedia.org/wiki/Quintic_function#Other_solvable_quintics

How to solve a cyclic quintic in radicals?

Solve this tough fifth degree equation.

$$ x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1 $$ $$ \Delta = 11^4 $$ $$ $$ $$ x^5 + x^4 - 12 x^3 - 21 x^2 + x + 5 $$ $$ \Delta = 5^2 \; 31^4 $$ $$ $$ $$ x^5 + x^4 - 16 x^3 + 5 x^2 + 21 x - 9 $$ $$ \Delta = 3^6 \; 41^4 $$ $$ $$ $$ x^5 + x^4 - 24 x^3 - 17 x^2 + 41 x - 13 $$ $$ \Delta = 29^2 \; 61^4 $$ $$ $$ $$ x^5 + x^4 - 28 x^3 + 37 x^2 + 25 x + 1 $$ $$ \Delta = 23^2 \; 71^4 $$ $$ $$ $$ x^5 + x^4 - 40 x^3 + 93 x^2 - 21 x - 17 $$ $$ \Delta = 17^2 \; 101^4 $$ $$ $$ $$ x^5 + x^4 - 52 x^3 - 89 x^2 + 109 x + 193 $$
$$ \Delta = 79^2 \; 131^4 $$ $$ $$ Tito(151) $$ x^5 + x^4 -60 x^3 -12 x^2 + 784 x + 128 $$ $$ \Delta = 2^{18} \; 151^4 $$ $$ $$ $$ x^5 + x^4 -72 x^3 -123 x^2 + 223 x -49 $$ $$ \Delta = 7^2 \; 149^2 \; 181^4 $$ $$ $$ Emma Lehmer(191) $$ x^5 + x^4 - 76 x^3 -359 x^2 - 437 x - 155 $$ $$ \Delta = 5^2 \; 11^2 \; 191^4 $$ $$ $$

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  • 1
    $\begingroup$ Given prime $p=10n+1$ and $$x^5+x^4-4nx^3+ax^2+bx+c=0$$ do you supposed we can also express $a,b,c$ in terms of $p$? It can be done for the cubic version as in this post but, of course, cubics are simpler. $\endgroup$ – Tito Piezas III Nov 20 '16 at 5:40
  • $\begingroup$ It seems if we "depress" these quintics (get rid of the $x^{n-1}$ term), we get the form, $$y^5+p(10y^3+ey^2+fy+g)=0$$ The discriminant $D$ of this form in general is $$D = p^4\,F(e,f,g)$$ but these cyclic quintics have $F(e,f,g) = z^2$. $\endgroup$ – Tito Piezas III Nov 21 '16 at 5:22
  • $\begingroup$ @Tito right. I put the general shape $x^5 + x^4 - 4 n x^3 + a x^2 + b x + c$ into gp-pari and asked for the discriminant. Then I edited the $a^2$ to $a * a$ and so on, finally put it into the Gauss quintic C++ program. The discriminant always comes out $z^2 p^4,$ so that the whole thing is a square. The process is quite deterministic, I imagine this is a theorem about the Gauss method, but I guess hideous to try to prove for the way I find them. $\endgroup$ – Will Jagy Nov 21 '16 at 17:06
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    $\begingroup$ I found some Diophantine relations between the quintics' coefficients. Kindly see this MO post. $\endgroup$ – Tito Piezas III Nov 23 '16 at 14:03
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    $\begingroup$ As an aside, "quintic with Galois group $\mathbb{Z}/5\mathbb{Z}$" automatically implies that the polynomial is irreducible and the roots are sums of roots of unity. $\endgroup$ – user14972 Dec 4 '16 at 21:00
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Yes, there are infinitely many cyclic quintics as parameterized by the Emma Lehmer quintic

$$F(y)=y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$

This also obeys

$$ y_1 y_2 + y_2 y_3 + y_3 y_4 + y_4 y_5 + y_5 y_1 - (y_1 y_3 + y_3 y_5 + y_5 y_2 + y_2 y_4 + y_4 y_1) = 0$$

Let $p=25 + 25 n + 15 n^2 + 5 n^3 + n^4$. Then the discriminant of $F(y)$ is

$$D = (7 + 10 n + 5 n^2 + n^3)^2\,p^4$$

Also, note that if $m=n+1$, then $n\,p=m^5 + 5m^3 + 5m - 11$. A root is given by $$y = a+b\sum_{k=1}^{(p-1)/5}\,{\zeta_p}^{c^k}$$

with root of unity $\zeta_p = e^{2\pi i/p},\,$ for some integer $a,b,c$. See this MO post for the formulas for $a,b,c$.

P.S. While $p=151$ does not belong to this family, I find that,

$$x^5 + x^4 - 60x^3 - 12x^2 + 784x + 128 = 0$$

with discriminant $d=2^{18}151^4$ has the root $\displaystyle x=\sum_{k=1}^{30}e^{2\pi\, i\, c^k/151}$ for $c=23$. The Mathematica command to find these quintics is,

Table[{c,Recognize[N[Sum[E^(2Pi I c^k/p),{k,1,(p-1)/5}],50],5,x]},{c,p/2}]

for prime $p\equiv1\pmod{10}$. Inspecting the resulting table of candidate quintics, identical ones with small coefficients will stand out and which gives the correct choice of $c$.

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  • $\begingroup$ A simple linear transformation can transform this family to the familiar form $$ x^5 + x^4 - 4 m x^3 + p x^2 + q x + r=0 $$ $\endgroup$ – Tito Piezas III Nov 3 '16 at 14:31
  • $\begingroup$ I programmed the method of Gauss, found in Galois Theory by D. A. Cox. Put answer with primes up to 311. $\endgroup$ – Will Jagy Nov 10 '16 at 22:46
  • $\begingroup$ oskicat.berkeley.edu/record=b16110183~S1 $\endgroup$ – Will Jagy Nov 10 '16 at 23:16
  • $\begingroup$ @TitoPiezasIII the polynomial that you write gives me for $m=0,p = 1,q = 0, r = 1$ Galois group $S_5$ in sage while I would expect it to still have galois group $C_5$ is that right? $\endgroup$ – Rodrigo Jan 4 '18 at 13:38
  • $\begingroup$ @Rodrigo: The variables $m,p,q,r$ are not independent, but are functions of the single variable $n$ used in the cyclic Emma Lehmer quintic in the post. $\endgroup$ – Tito Piezas III Jan 4 '18 at 13:53
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There is also an infinite number of cyclic septics, such as the Hashimoto-Hoshi, $$\small x^7 - (a^3 + a^2 + 5a + 6)x^6 + 3(3a^3 + 3a^2 + 8a + 4)x^5 + (a^7 + a^6 + 9a^5 - 5a^4 - 15a^3 - 22a^2 - 36a - 8)x^4 - a(a^7 + 5a^6 + 12a^5 + 24a^4 - 6a^3 + 2a^2 - 20a - 16)x^3 + a^2(2a^6 + 7a^5 + 19a^4 + 14a^3 + 2a^2 + 8a - 8)x^2 - a^4(a^4 + 4a^3 + 8a^2 + 4)x + a^7=0$$

Similar to the Lehmer quintic, the roots of this septic obeys $$ x_1 x_2 + x_2 x_3 + \dots + x_7 x_1 - (x_1 x_3 + x_3 x_5 + \dots + x_6 x_1) = 0$$ For example, let $a=1$ so, $$1 - 17 x + 44 x^2 - 2 x^3 - 75 x^4 + 54 x^5 - 13 x^6 + x^7=0$$ which is the equation involved in $\cos\frac{\pi k}{43}$. See also this post On solvable quintics and septics.

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  • $\begingroup$ Probably take a few days, I can make the Gauss method for degree seven, for primes $p \equiv 1 \pmod 7.$ I see, if you took $x = t + 2$ in you 43 example, you would get $t^7 + t^6...$ $\endgroup$ – Will Jagy Nov 25 '16 at 18:15
  • $\begingroup$ I managed to push together something to, slowly, compute the septics by Gauss. I think these are not the same as your family. Also getting messier as the number of unknown coefficients increases. $\endgroup$ – Will Jagy Nov 28 '16 at 1:02
  • $\begingroup$ @WillJagy: Just like there's one super-family of such quintics as discussed in that MO post, there's probably a super-family of such septics as well. $\endgroup$ – Tito Piezas III Nov 28 '16 at 1:23
  • $\begingroup$ OK. Well, the method of Gauss gives an example for every prime $p \equiv 1 \pmod q,$ where $q$ is the degree of the polynomial and is prime as well. However, a bit of work to construct them. Oh, I did borrow the Storer book, Cyclotomy and Difference Sets. I think it is all applicable, but he does not seem to write any of these polynomials. $\endgroup$ – Will Jagy Nov 28 '16 at 2:22
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The method of Gauss for this problem is presented in Chapter 9 of Galois Theory by David A. Cox. This was worked out about 30 years before Galois Theory. After doing primes 31, 61, 71 by hand as illustrated there, I was able to write a straightforward program in C++. The input is the prime $p = 10 n + 1$ and a primitive root for that prime. I could have just told the computer to find a primitive root, since I intended primes smaller than 1000 in any case. As I did more of them, I had the machine give a better output; still, for all of these, you will be able to read the quintic polynomial and the collection of exponents of the original $\zeta = e^{2 \pi i / p};$ the sum of these $\zeta^k$ gives one of the five real roots.

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  sofar         1          1          1          1          1          1


  prime was  11

  primitive root used was  2

 smallest generator  is 10

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1

 list of the 2 exponents 
       1      10

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  sofar       185        185        185        185        185        185


  prime was  31

  primitive root used was  3

 smallest generator  is 6

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 12 x^3 - 21 x^2 + 1 x + 5

 list of the 6 exponents 
       1       5       6      25      26      30

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  sofar       711        711        711        711        711        711


  prime was  41

  primitive root used was  6

 smallest generator  is 3

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 16 x^3 + 5 x^2 + 21 x - 9

 list of the 8 exponents 
       1       3       9      14      27      32      38      40

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sofar      3707       3707       3707       3707       3707       3707


  prime was  61

  primitive root used was  2

 smallest generator  is 21

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 24 x^3 - 17 x^2 + 41 x - 13

 list of the 12 exponents 
       1      11      13      14      21      29      32      40      47      48
      50      60

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  sofar      7141       7141       7141       7141       7141       7141


  prime was  71

  primitive root used was  7

 smallest generator  is 23

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 28 x^3 + 37 x^2 + 25 x + 1

 list of the 14 exponents 
       1      20      23      26      30      32      34      37      39      41
      45      48      51      70

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  sofar     30463      30463      30463      30463      30463      30463


  prime was  101

  primitive root used was  2

 smallest generator  is 32

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 40 x^3 + 93 x^2 - 21 x - 17

 list of the 20 exponents 
       1       6      10      14      17      32      36      39      41      44
      57      60      62      65      69      84      87      91      95     100

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  sofar     86773      86773      86773      86773      86773      86773


  prime was  131

  primitive root used was  2

 smallest generator  is 18

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 52 x^3 - 89 x^2 + 109 x + 193

 list of the 26 exponents 
       1      18      19      24      32      39      45      47      51      52
      60      62      63      68      69      71      79      80      84      86
      92      99     107     112     113     130

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  sofar    155648     155648     155648     155648     155648     155648


  prime was  151

  primitive root used was  6

 smallest generator  is 23

 actual value of the constant a, usually 2 but not always, was  3

polynomial is  x^5 + x^4 - 60 x^3 - 12 x^2 + 784 x + 128

 list of the 30 exponents 
       1       2       4       8      16      19      23      32      33      38
      46      59      64      66      75      76      85      87      92     105
     113     118     119     128     132     135     143     147     149     150

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  sofar    323951     323951     323951     323951     323951     323951


  prime was  181

  primitive root used was  2

 smallest generator  is 17

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 72 x^3 - 123 x^2 + 223 x - 49

 list of the 36 exponents 
       1       7      17      19      26      32      39      43      48      49
      61      62      65      72      73      80      88      89      92      93
     101     108     109     116     119     120     132     133     138     142
     149     155     162     164     174     180

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  sofar    401125     401125     401125     401125     401125     401125


  prime was  191

  primitive root used was  19

 smallest generator  is 11

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 76 x^3 - 359 x^2 - 437 x - 155

 list of the 38 exponents 
       1       5       6      11      14      25      30      31      32      36
      37      38      41      52      55      66      69      70      84     107
     121     122     125     136     139     150     153     154     155     159
     160     161     166     177     180     185     186     190

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  sofar    604481     604481     604481     604481     604481     604481


  prime was  211

  primitive root used was  2

 smallest generator  is 26

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 84 x^3 - 59 x^2 + 1661 x + 269

 list of the 42 exponents 
       1      12      14      15      26      31      32      33      34      38
      40      43      50      54      58      63      67      73      88      94
     101     110     117     123     138     144     148     153     157     161
     168     171     173     177     178     179     180     185     196     197
     199     210

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  sofar   1033472    1033472    1033472    1033472    1033472    1033472


  prime was  241

  primitive root used was  7

 smallest generator  is 11

 actual value of the constant a, usually 2 but not always, was  3

polynomial is  x^5 + x^4 - 96 x^3 - 212 x^2 + 1232 x + 512

 list of the 48 exponents 
       1       2       4       8      11      15      16      19      22      30
      32      38      44      60      63      64      65      76      88      89
     111     113     115     120     121     126     128     130     152     153
     165     176     177     178     181     197     203     209     211     219
     222     225     226     230     233     237     239     240

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  sofar   1220224    1220224    1220224    1220224    1220224    1220224


  prime was  251

  primitive root used was  6

 smallest generator  is 2

 actual value of the constant a, usually 2 but not always, was  3

polynomial is  x^5 + x^4 - 100 x^3 - 20 x^2 + 1504 x + 1024

 list of the 50 exponents 
       1       2       4       5       8      10      16      20      25      32
      40      47      50      51      63      64      69      80      91      94
     100     102     113     123     125     126     128     138     149     151
     157     160     171     182     187     188     200     201     204     211
     219     226     231     235     241     243     246     247     249     250

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  sofar   1658645    1658645    1658645    1658645    1658645    1658645


  prime was  271

  primitive root used was  6

 smallest generator  is 12

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 108 x^3 - 401 x^2 - 13 x + 845

 list of the 54 exponents 
       1       5      12      13      23      25      28      29      32      33
      54      60      65      77      83      88      93     102     106     111
     113     114     115     125     126     127     131     140     144     145
     146     156     157     158     160     165     169     178     183     188
     194     206     211     217     238     239     242     243     246     248
     258     259     266     270

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar   1923223    1923223    1923223    1923223    1923223    1923223


  prime was  281

  primitive root used was  3

 smallest generator  is 6

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 112 x^3 - 191 x^2 + 2257 x + 967

 list of the 56 exponents 
       1       6      10      28      32      34      36      37      38      39
      47      53      59      60      65      73      77      79      88      89
      92      99     100     109     113     116     124     134     147     157
     165     168     172     181     182     189     192     193     202     204
     208     216     221     222     228     234     242     243     244     245
     247     249     253     271     275     280

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar   2904781    2904781    2904781    2904781    2904781    2904781


  prime was  311

  primitive root used was  17

 smallest generator  is 11

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 124 x^3 + 535 x^2 - 413 x - 539

 list of the 62 exponents 
       1       7      11      13      15      18      20      24      32      41
      46      47      49      51      61      68      77      83      86      87
      89      91     105     113     116     121     126     140     142     143
     146     165     168     169     171     185     190     195     198     206
     220     222     224     225     228     234     243     250     260     262
     264     265     270     279     287     291     293     296     298     300
     304     310

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar   3714113    3714113    3714113    3714113    3714113    3714113


  prime was  331

  primitive root used was  3

 smallest generator  is 13

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 132 x^3 - 887 x^2 - 1843 x - 1027

 list of the 66 exponents 
       1      13      23      31      32      34      38      47      48      51
      57      61      72      74      79      80      85      88      89      95
     108     111     112     119     120     131     132     133     146     151
     162     163     164     167     168     169     180     185     198     199
     200     211     212     219     220     223     236     242     243     246
     251     252     257     259     270     274     280     283     284     293
     297     299     300     308     318     330

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar   8075491    8075491    8075491    8075491    8075491    8075491


  prime was  401

  primitive root used was  3

 smallest generator  is 26

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 160 x^3 + 369 x^2 + 879 x - 29

 list of the 80 exponents 
       1      20      22      26      29      30      32      33      35      39
      45      48      56      68      72      76      83      84      98     102
     108     114     119     126     133     142     147     148     151     153
     155     157     158     162     164     171     179     188     189     199
     202     212     213     222     230     237     239     243     244     246
     248     250     253     254     259     268     275     282     287     293
     299     303     317     318     325     329     333     345     353     356
     362     366     368     369     371     372     375     379     381     400

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar   9819947    9819947    9819947    9819947    9819947    9819947


  prime was  421

  primitive root used was  2

 smallest generator  is 32

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 168 x^3 + 219 x^2 + 3853 x - 3517

 list of the 84 exponents 
       1       6      20      21      29      32      33      36      51      52
      70      75      86      93      95     109     110     111     112     115
     120     122     126     135     137     149     152     159     170     171
     174     176     178     182     184     188     192     195     198     202
     205     207     214     216     219     223     226     229     233     237
     239     243     245     247     250     251     262     269     272     284
     286     295     299     301     306     309     310     311     312     326
     328     335     346     351     369     370     385     388     389     392
     400     401     415     420

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar  14139931   14139931   14139931   14139931   14139931   14139931


  prime was  461

  primitive root used was  2

 smallest generator  is 13

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 184 x^3 - 129 x^2 + 4551 x + 5419

 list of the 92 exponents 
       1      13      14      20      21      22      23      30      32      33
      37      38      41      45      48      57      61      68      71      72
      86     102     108     113     124     129     134     139     145     153
     162     163     167     169     175     179     181     182     186     188
     196     199     201     211     218     229     232     243     250     260
     262     265     273     275     279     280     282     286     292     294
     298     299     308     316     322     327     332     337     348     353
     359     375     389     390     393     400     404     413     416     420
     423     424     428     429     431     438     439     440     441     447
     448     460

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar  18223497   18223497   18223497   18223497   18223497   18223497


  prime was  491

  primitive root used was  2

 smallest generator  is 32

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 196 x^3 + 59 x^2 + 2019 x + 1377

 list of the 98 exponents 
       1       3       9      14      17      27      32      35      37      42
      43      46      51      53      77      80      81      96      97     104
     105     109     111     113     115     118     126     129     137     138
     146     152     153     158     159     164     176     178     179     196
     200     202     203     223     229     231     238     240     243     248
     251     253     260     262     268     288     289     291     295     312
     313     315     327     332     333     338     339     345     353     354
     362     365     373     376     378     380     382     386     387     394
     395     410     411     414     438     440     445     448     449     454
     456     459     464     474     477     482     488     490

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

  sofar  23112547   23112547   23112547   23112547   23112547   23112547


  prime was  521

  primitive root used was  3

 smallest generator  is 24

 actual value of the constant a, usually 2 but not always, was  2

polynomial is  x^5 + x^4 - 208 x^3 - 771 x^2 + 4143 x + 2083

 list of the 104 exponents 
       1      10      18      24      29      32      34      39      42      43
      46      52      55      56      61      62      74      75      89      91
      98      99     100     101     106     114     131     132     135     152
     175     176     180     181     187     197     201     205     206     213
     214     219     226     229     231     235     237     240     243     247
     253     255     266     268     274     278     281     284     286     290
     292     295     302     307     308     315     316     320     324     334
     340     341     345     346     369     386     389     390     407     415
     420     421     422     423     430     432     446     447     459     460
     465     466     469     475     478     479     482     487     489     492
     497     503     511     520

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
$\endgroup$
0
$\begingroup$

DEGREE $7$

As a separate item, for prime polynomial degree we do get an example with one root being the sum of $2 \cos \frac{2 \pi m^k}{p^2},$ here $p^2 = 49$

$$ X^7 -21 X^5 -21 X^4 + 91 X^3 + 112 X^2 -84 X -97, \;\; 19^k $$

Let's see, below $r$ is the primitive root for $p$ used, then the list of exponents (of $e^{2 \pi i / p}$) is the sum of powers of the given figure, 12 or 7 or 14 or 35. The first one and its roots are

parisize = 4000000, primelimit = 500509
? f = x^7 + x^6 - 12 * x^5 - 7 * x^4 + 28 * x^3 + 14 * x^2 - 9 * x + 1
%1 = x^7 + x^6 - 12*x^5 - 7*x^4 + 28*x^3 + 14*x^2 - 9*x + 1
? polroots(f)
%2 = [
-3.347297326211866604824677822 + 0.E-28*I, 
-1.453219237250277575521353021 + 0.E-28*I, 
-1.063840303785358166816481464 + 0.E-28*I, 
 0.1723984388388905398234384116 + 0.E-28*I, 
 0.2395267590849948773703028220 + 0.E-28*I, 
 1.700463948582122544295969145 + 0.E-28*I, 
 2.751967720741494385672801928 + 0.E-28*I]~
? 

or as sums of cosines $$ 2 \cos \left( \frac{2 \pi}{29} \right) + 2 \cos \left( \frac{24 \pi}{29} \right) = 2 \cos \left( \frac{2 \pi}{29} \right) - 2 \cos \left( \frac{5 \pi}{29} \right) \approx 0.239526759 $$ $$ 2 \cos \left( \frac{4 \pi}{29} \right) + 2 \cos \left( \frac{48 \pi}{29} \right) = 2 \cos \left( \frac{4 \pi}{29} \right) + 2 \cos \left( \frac{10 \pi}{29} \right) \approx 2.75196772 $$ $$ 2 \cos \left( \frac{8 \pi}{29} \right) + 2 \cos \left( \frac{96 \pi}{29} \right) = 2 \cos \left( \frac{8 \pi}{29} \right) - 2 \cos \left( \frac{9 \pi}{29} \right) \approx 0.1723984 $$ $$ 2 \cos \left( \frac{16 \pi}{29} \right) + 2 \cos \left( \frac{192 \pi}{29} \right) = -2 \cos \left( \frac{13 \pi}{29} \right) - 2 \cos \left( \frac{11 \pi}{29} \right) \approx -1.06384 $$ $$ 2 \cos \left( \frac{32 \pi}{29} \right) + 2 \cos \left( \frac{384 \pi}{29} \right) = -2 \cos \left( \frac{3 \pi}{29} \right) - 2 \cos \left( \frac{7 \pi}{29} \right) \approx -3.347297326 $$ $$ 2 \cos \left( \frac{64 \pi}{29} \right) + 2 \cos \left( \frac{768 \pi}{29} \right) = 2 \cos \left( \frac{6 \pi}{29} \right) + 2 \cos \left( \frac{14 \pi}{29} \right) \approx 1.70046 $$ $$ 2 \cos \left( \frac{128 \pi}{29} \right) + 2 \cos \left( \frac{1536 \pi}{29} \right) = 2 \cos \left( \frac{12 \pi}{29} \right) - 2 \cos \left( \frac{ \pi}{29} \right) \approx -1.4532 $$

The first three below are in Reuschle pages 35, 66, 113.

$$ x^7 + x^6 - 12 x^5 - 7 x^4 + 28 x^3 + 14 x^2 - 9 x + 1, \; \; p = 29, \; \; r = 2, \; \; 12^k $$ $$ x^7 + x^6 - 18 x^5 - 35 x^4 + 38 x^3 + 104 x^2 + 7 x - 49, \; \; p = 43, \; \; r = 3, \; \; 7^k $$ $$ x^7 + x^6 - 30 x^5 + 3 x^4 + 254 x^3 - 246 x^2 - 245 x + 137, \; \; p = 71, \; \; r = 7, \; \; 14^k $$ $$ x^7 + x^6 - 48 x^5 + 37 x^4 + 312 x^3 - 12 x^2 - 49 x - 1, \; \; p = 113, \; \; r = 3, \; \; 35^k $$
$$ x^7 + x^6 - 54 x^5 - 31 x^4 + 558 x^3 - 32 x^2 - 1713 x + 1121, \; \; p = 127, \; \; r = 3, \; \; 24^k $$ $$ x^7 + x^6 - 84 x^5 - 217 x^4 + 1348 x^3 + 3988 x^2 - 1433 x - 1163, \; \; p = 197, \; \; r = 2, \; \; 20^k $$ $$ x^7 + x^6 - 90 x^5 + 69 x^4 + 1306 x^3 + 124 x^2 - 5249 x - 4663, \; \; p = 211, \; \; r = 2, \; \; 10^k $$ $$ x^7 + x^6 - 102 x^5 - 195 x^4 + 1850 x^3 + 978 x^2 - 8933 x + 5183, \; \; p = 239, \; \; r = 7, \; \; 23^k $$ $$ x^7 + x^6 - 120 x^5 - 711 x^4 - 784 x^3 + 1956 x^2 + 2863 x - 343, \; \; p = 281, \; \; r = 3, \; \; 61^k $$ $$ x^7 + x^6 - 144 x^5 + 399 x^4 + 2416 x^3 - 10808 x^2 + 10831 x - 1237, \; \; p = 337, \; \; r = 10, \; \; 38^k $$ $$ x^7 + x^6 -162 x^5 -201 x^4 + 7822 x^3 + 12322 x^2 -107717 x -193369, \; \; p= 379, \; \; 11^k $$
$$ x^7 + x^6 -180 x^5 -103 x^4 + 6180 x^3 + 11596 x^2 -25209 x -49213, \; \; p= 421, \; \; 34^k $$ $$ x^7 + x^6 -192 x^5 + 275 x^4 + 3952 x^3 + 4136 x^2 -81 x -863, \; \; p= 449, \; \; 24^k $$ $$ x^7 + x^6 -198 x^5 -907 x^4 + 4302 x^3 + 20582 x^2 -18973 x -56911, \; \; p= 463, \; \; 6^k $$ $$ x^7 + x^6 -210 x^5 + 1423 x^4 -1410 x^3 -8538 x^2 + 9203 x + 19427, \; \; p =491, \; \; 63^k $$ $$ x^7 + x^6 -234 x^5 + 335 x^4 + 13254 x^3 -42874 x^2 -55309 x + 71879, \; \; p=547, \; \; 26^k $$

enter image description here enter image description here enter image description here

      x^7 + x^6 - 12 x^5 + -7 x^4 + 28 x^3 + 14 x^2 + -9 x + 1
       p  29 p.root  2 exps 12^k 
     list of the 4 exponents 
           1      12      17      28
    ==============================================
      x^7 + x^6 - 18 x^5 + -35 x^4 + 38 x^3 + 104 x^2 + 7 x + -49
       p  43 p.root  3 exps 7^k 
     list of the 6 exponents 
           1       6       7      36      37      42  
    ==========================================================
      x^7 + x^6 - 30 x^5 + 3 x^4 + 254 x^3 + -246 x^2 + -245 x + 137
       p  71 p.root  7 exps 14^k 
     list of the 10 exponents 
           1       5      14      17      25      46      54      57      66      70
   ====================================================
  x^7 + x^6 - 48 x^5 + 37 x^4 + 312 x^3 + -12 x^2 + -49 x + -1
   p  113 p.root  3 exps 35^k 
 list of the 16 exponents 
       1      15      18      35      40      42      44      48      65      69
      71      73      78      95      98     112
$\endgroup$

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