# Any more cyclic quintics?

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 ===============================================================================================================  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) $$x^5 + x^4 - 76 x^3 -359 x^2 - 437 x - 155$$ $$\Delta = 5^2 \; 11^2 \; 191^4$$  =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= • 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. – Tito Piezas III Nov 20 '16 at 5:40 • 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$F(e,f,g) = z^2$. – Tito Piezas III Nov 21 '16 at 5:22 • @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. – Will Jagy Nov 21 '16 at 17:06 • I found some Diophantine relations between the quintics' coefficients. Kindly see this MO post. – Tito Piezas III Nov 23 '16 at 14:03 • 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. – user14972 Dec 4 '16 at 21:00 ## 4 Answers 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$c$. • A simple linear transformation can transform this family to the familiar form $$x^5 + x^4 - 4 m x^3 + p x^2 + q x + r=0$$ – Tito Piezas III Nov 3 '16 at 14:31 • I programmed the method of Gauss, found in Galois Theory by D. A. Cox. Put answer with primes up to 311. – Will Jagy Nov 10 '16 at 22:46 • oskicat.berkeley.edu/record=b16110183~S1 – Will Jagy Nov 10 '16 at 23:16 • @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? – Rodrigo Jan 4 '18 at 13:38 • @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. – Tito Piezas III Jan 4 '18 at 13:53 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. • 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$t^7 + t^6...$– Will Jagy Nov 25 '16 at 18:15 • 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. – Will Jagy Nov 28 '16 at 1:02 • @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. – Tito Piezas III Nov 28 '16 at 1:23 • 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. – Will Jagy Nov 28 '16 at 2:22 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. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= 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 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=  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.

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

      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