Gaussian period polynomials

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Polynomials

$p$

$k$

$\Psi_{p,k}(x)$

x^2 + x - 1

comment: $p^*=5$; roots $\frac{-1\pm\sqrt{5}}{2}$, in Algebraic numbers of degree 2; $f=2$: the roots are $2\cos(2\pi a/5)$, $1\leq a\leq 2$

x^2 + x + 2

comment: $p^*=-7$; roots $\frac{-1\pm\sqrt{-7}}{2}$, in Algebraic numbers of degree 2

x^3 + x^2 - 2*x - 1

comment: $4p=L^2+27M^2$ with $L=1$, $M=1$; $f=2$: the roots are $2\cos(2\pi a/7)$, $1\leq a\leq 3$

x^2 + x + 3

comment: $p^*=-11$; roots $\frac{-1\pm\sqrt{-11}}{2}$, in Algebraic numbers of degree 2

x^5 + x^4 - 4*x^3 - 3*x^2 + 3*x + 1

comment: $f=2$: the roots are $2\cos(2\pi a/11)$, $1\leq a\leq 5$

x^2 + x - 3

comment: $p^*=13$; roots $\frac{-1\pm\sqrt{13}}{2}$, in Algebraic numbers of degree 2

x^3 + x^2 - 4*x + 1

comment: $4p=L^2+27M^2$ with $L=-5$, $M=1$

x^4 + x^3 + 2*x^2 - 4*x + 3

x^6 + x^5 - 5*x^4 - 4*x^3 + 6*x^2 + 3*x - 1

comment: $f=2$: the roots are $2\cos(2\pi a/13)$, $1\leq a\leq 6$

x^2 + x - 4

comment: $p^*=17$; roots $\frac{-1\pm\sqrt{17}}{2}$, in Algebraic numbers of degree 2; a period equation of Gauss's construction of the regular 17-gon

x^4 + x^3 - 6*x^2 - x + 1

comment: a period equation of Gauss's construction of the regular 17-gon

x^8 + x^7 - 7*x^6 - 6*x^5 + 15*x^4 + 10*x^3 - 10*x^2 - 4*x + 1

comment: $f=2$: the roots are $2\cos(2\pi a/17)$, $1\leq a\leq 8$; a period equation of Gauss's construction of the regular 17-gon

x^2 + x + 5

comment: $p^*=-19$; roots $\frac{-1\pm\sqrt{-19}}{2}$, in Algebraic numbers of degree 2

x^3 + x^2 - 6*x - 7

comment: $4p=L^2+27M^2$ with $L=7$, $M=1$

x^6 + x^5 + 2*x^4 - 8*x^3 - x^2 + 5*x + 7

x^9 + x^8 - 8*x^7 - 7*x^6 + 21*x^5 + 15*x^4 - 20*x^3 - 10*x^2 + 5*x + 1

comment: $f=2$: the roots are $2\cos(2\pi a/19)$, $1\leq a\leq 9$

x^2 + x + 6

comment: $p^*=-23$; roots $\frac{-1\pm\sqrt{-23}}{2}$

11:

x^11 + x^10 - 10*x^9 - 9*x^8 + 36*x^7 + 28*x^6 - 56*x^5 - 35*x^4 + 35*x^3 + 15*x^2 - 6*x - 1

comment: $f=2$: the roots are $2\cos(2\pi a/23)$, $1\leq a\leq 11$

x^2 + x - 7

comment: $p^*=29$; roots $\frac{-1\pm\sqrt{29}}{2}$

x^4 + x^3 + 4*x^2 + 20*x + 23

x^7 + x^6 - 12*x^5 - 7*x^4 + 28*x^3 + 14*x^2 - 9*x + 1

x^2 + x + 8

comment: $p^*=-31$; roots $\frac{-1\pm\sqrt{-31}}{2}$

x^3 + x^2 - 10*x - 8

comment: $4p=L^2+27M^2$ with $L=4$, $M=2$

x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

x^6 + x^5 + 3*x^4 + 11*x^3 + 44*x^2 + 36*x + 32

10:

x^10 + x^9 + 2*x^8 - 16*x^7 - 9*x^6 - 11*x^5 + 43*x^4 + 6*x^3 + 63*x^2 + 20*x + 25

x^2 + x - 9

comment: $p^*=37$; roots $\frac{-1\pm\sqrt{37}}{2}$

x^3 + x^2 - 12*x + 11

comment: $4p=L^2+27M^2$ with $L=-11$, $M=1$

x^4 + x^3 + 5*x^2 + 7*x + 49

x^6 + x^5 - 15*x^4 - 28*x^3 + 15*x^2 + 38*x - 1

x^9 + x^8 - 16*x^7 - 11*x^6 + 66*x^5 + 32*x^4 - 73*x^3 - 7*x^2 + 7*x + 1

12:

x^12 + x^11 + 2*x^10 - 20*x^9 - 13*x^8 - 19*x^7 + 85*x^6 + 51*x^5 + 94*x^4 - 2*x^3 - 13*x^2 - 77*x + 47

x^2 + x - 10

comment: $p^*=41$; roots $\frac{-1\pm\sqrt{41}}{2}$

x^4 + x^3 - 15*x^2 + 18*x - 4

x^5 + x^4 - 16*x^3 + 5*x^2 + 21*x - 9

x^8 + x^7 + 3*x^6 + 11*x^5 + 44*x^4 - 53*x^3 + 153*x^2 - 160*x + 59

10:

x^10 + x^9 - 18*x^8 - 13*x^7 + 91*x^6 + 47*x^5 - 143*x^4 - 7*x^3 + 72*x^2 - 23*x + 1

x^2 + x + 11

comment: $p^*=-43$; roots $\frac{-1\pm\sqrt{-43}}{2}$

x^3 + x^2 - 14*x + 8

comment: $4p=L^2+27M^2$ with $L=-8$, $M=2$

x^6 + x^5 + 4*x^4 - 23*x^3 + 67*x^2 - 50*x + 44

x^7 + x^6 - 18*x^5 - 35*x^4 + 38*x^3 + 104*x^2 + 7*x - 49

x^2 + x + 12

comment: $p^*=-47$; roots $\frac{-1\pm\sqrt{-47}}{2}$

x^2 + x - 13

comment: $p^*=53$; roots $\frac{-1\pm\sqrt{53}}{2}$

x^4 + x^3 + 7*x^2 - 43*x + 47

x^2 + x + 15

comment: $p^*=-59$; roots $\frac{-1\pm\sqrt{-59}}{2}$

x^2 + x - 15

comment: $p^*=61$; roots $\frac{-1\pm\sqrt{61}}{2}$

x^3 + x^2 - 20*x - 9

comment: $4p=L^2+27M^2$ with $L=1$, $M=3$

x^4 + x^3 + 8*x^2 + 42*x + 117

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

x^6 + x^5 - 25*x^4 + 8*x^3 + 123*x^2 - 126*x + 27

10:

x^10 + x^9 - 27*x^8 - 56*x^7 + 161*x^6 + 500*x^5 + x^4 - 1023*x^3 - 916*x^2 - 202*x - 13

12:

x^12 + x^11 + 3*x^10 + 11*x^9 - 17*x^8 - 169*x^7 + 325*x^6 + 167*x^5 - 804*x^4 + 160*x^3 + 1102*x^2 - 780*x + 1179

x^2 + x + 17

comment: $p^*=-67$; roots $\frac{-1\pm\sqrt{-67}}{2}$

x^3 + x^2 - 22*x + 5

comment: $4p=L^2+27M^2$ with $L=-5$, $M=3$

x^6 + x^5 + 6*x^4 + 46*x^3 + 123*x^2 + 169*x + 617

11:

x^11 + x^10 - 30*x^9 - 63*x^8 + 220*x^7 + 698*x^6 - 101*x^5 - 1960*x^4 - 1758*x^3 - 35*x^2 + 243*x - 29

x^2 + x + 18

comment: $p^*=-71$; roots $\frac{-1\pm\sqrt{-71}}{2}$

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

x^7 + x^6 - 30*x^5 + 3*x^4 + 254*x^3 - 246*x^2 - 245*x + 137

10:

x^10 + x^9 + 4*x^8 + 20*x^7 - 103*x^6 + 141*x^5 + 207*x^4 - 1254*x^3 + 2635*x^2 - 4020*x + 3737

x^2 + x - 18

comment: $p^*=73$; roots $\frac{-1\pm\sqrt{73}}{2}$

x^3 + x^2 - 24*x - 27

comment: $4p=L^2+27M^2$ with $L=7$, $M=3$

x^4 + x^3 - 27*x^2 - 41*x + 2

x^6 + x^5 - 30*x^4 - 31*x^3 + 206*x^2 + 150*x - 81

x^8 + x^7 + 5*x^6 - 17*x^5 - 46*x^4 - 136*x^3 + 320*x^2 + 512*x + 4096

x^9 + x^8 - 32*x^7 - 11*x^6 + 278*x^5 - 34*x^4 - 427*x^3 + 150*x^2 - 8*x - 1

12:

x^12 + x^11 - 33*x^10 - 70*x^9 + 288*x^8 + 929*x^7 - 298*x^6 - 3421*x^5 - 2921*x^4 + 1195*x^3 + 1718*x^2 - 162*x - 211

x^2 + x + 20

comment: $p^*=-79$; roots $\frac{-1\pm\sqrt{-79}}{2}$

x^3 + x^2 - 26*x + 41

comment: $4p=L^2+27M^2$ with $L=-17$, $M=1$

x^6 + x^5 + 7*x^4 + 63*x^3 - 81*x^2 - 353*x + 541

x^2 + x + 21

comment: $p^*=-83$; roots $\frac{-1\pm\sqrt{-83}}{2}$

x^2 + x - 22

comment: $p^*=89$; roots $\frac{-1\pm\sqrt{89}}{2}$

x^4 + x^3 - 33*x^2 + 39*x + 8

x^8 + x^7 + 6*x^6 + 46*x^5 - 143*x^4 - 575*x^3 + 1160*x^2 + 16*x + 512

11:

x^11 + x^10 - 40*x^9 - 19*x^8 + 482*x^7 + 84*x^6 - 2185*x^5 + 102*x^4 + 3152*x^3 - 781*x^2 + 57*x - 1

x^2 + x - 24

comment: $p^*=97$; roots $\frac{-1\pm\sqrt{97}}{2}$

x^3 + x^2 - 32*x - 79

comment: $4p=L^2+27M^2$ with $L=19$, $M=1$

x^4 + x^3 - 36*x^2 + 91*x - 61

x^6 + x^5 - 40*x^4 + 45*x^3 + 236*x^2 - 230*x - 389

x^8 + x^7 - 42*x^6 - 59*x^5 + 497*x^4 + 719*x^3 - 1792*x^2 - 2295*x + 193

12:

x^12 + x^11 - 44*x^10 - 23*x^9 + 608*x^8 + 288*x^7 - 3367*x^6 - 1647*x^5 + 7459*x^4 + 2633*x^3 - 7037*x^2 - 1034*x + 2209

101

x^2 + x - 25

comment: $p^*=101$; roots $\frac{-1\pm\sqrt{101}}{2}$

101

x^4 + x^3 + 13*x^2 + 19*x + 361

101

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

101

10:

x^10 + x^9 - 45*x^8 - 12*x^7 + 614*x^6 - 399*x^5 - 2937*x^4 + 3927*x^3 + 3176*x^2 - 7776*x + 3433

103

x^2 + x + 26

comment: $p^*=-103$; roots $\frac{-1\pm\sqrt{-103}}{2}$

103

x^3 + x^2 - 34*x - 61

comment: $4p=L^2+27M^2$ with $L=13$, $M=3$

103

x^6 + x^5 + 9*x^4 - 101*x^3 + 129*x^2 + 91*x + 1373

107

x^2 + x + 27

comment: $p^*=-107$; roots $\frac{-1\pm\sqrt{-107}}{2}$

109

x^2 + x - 27

comment: $p^*=109$; roots $\frac{-1\pm\sqrt{109}}{2}$

109

x^3 + x^2 - 36*x - 4

comment: $4p=L^2+27M^2$ with $L=-2$, $M=4$

109

x^4 + x^3 + 14*x^2 - 34*x + 393

109

x^6 + x^5 - 45*x^4 - 10*x^3 + 135*x^2 + 9*x - 27

109

x^9 + x^8 - 48*x^7 - 73*x^6 + 660*x^5 + 1454*x^4 - 2149*x^3 - 8350*x^2 - 7432*x - 2008

109

12:

x^12 + x^11 + 5*x^10 - 41*x^9 + 149*x^8 - 794*x^7 + 2669*x^6 + 759*x^5 + 3238*x^4 + 2526*x^3 + 13868*x^2 + 6509*x + 2503

113

x^2 + x - 28

comment: $p^*=113$; roots $\frac{-1\pm\sqrt{113}}{2}$

113

x^4 + x^3 - 42*x^2 - 120*x - 64

113

x^7 + x^6 - 48*x^5 + 37*x^4 + 312*x^3 - 12*x^2 - 49*x - 1

113

x^8 + x^7 - 49*x^6 + 16*x^5 + 511*x^4 - 367*x^3 - 1499*x^2 + 798*x + 1372

127

x^2 + x + 32

comment: $p^*=-127$; roots $\frac{-1\pm\sqrt{-127}}{2}$

127

x^3 + x^2 - 42*x + 80

comment: $4p=L^2+27M^2$ with $L=-20$, $M=2$

127

x^6 + x^5 + 11*x^4 - 181*x^3 + 660*x^2 - 972*x + 608

127

x^7 + x^6 - 54*x^5 - 31*x^4 + 558*x^3 - 32*x^2 - 1713*x + 1121

127

x^9 + x^8 - 56*x^7 - 118*x^6 + 573*x^5 + 1249*x^4 - 1582*x^3 - 2700*x^2 + 1576*x + 32

131

x^2 + x + 33

comment: $p^*=-131$; roots $\frac{-1\pm\sqrt{-131}}{2}$

131

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

131

10:

x^10 + x^9 + 7*x^8 + 63*x^7 + 237*x^6 + 783*x^5 + 7565*x^4 + 21935*x^3 + 39574*x^2 + 36034*x + 18289

137

x^2 + x - 34

comment: $p^*=137$; roots $\frac{-1\pm\sqrt{137}}{2}$

137

x^4 + x^3 - 51*x^2 - 214*x - 236

137

x^8 + x^7 + 9*x^6 + 105*x^5 + 954*x^4 + 3767*x^3 + 9149*x^2 + 12828*x + 7607

139

x^2 + x + 35

comment: $p^*=-139$; roots $\frac{-1\pm\sqrt{-139}}{2}$

139

x^3 + x^2 - 46*x + 103

comment: $4p=L^2+27M^2$ with $L=-23$, $M=1$

139

x^6 + x^5 + 12*x^4 + 188*x^3 - 46*x^2 - 1356*x + 1723

149

x^2 + x - 37

comment: $p^*=149$; roots $\frac{-1\pm\sqrt{149}}{2}$

149

x^4 + x^3 + 19*x^2 - 121*x + 635

151

x^2 + x + 38

comment: $p^*=-151$; roots $\frac{-1\pm\sqrt{-151}}{2}$

151

x^3 + x^2 - 50*x - 123

comment: $4p=L^2+27M^2$ with $L=19$, $M=3$

151

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

151

x^6 + x^5 + 13*x^4 - 81*x^3 - 331*x^2 + 1347*x + 6543

151

10:

x^10 + x^9 + 8*x^8 - 18*x^7 + 397*x^6 - 351*x^5 + 4010*x^4 - 720*x^3 + 4352*x^2 + 11264*x + 292352

157

x^2 + x - 39

comment: $p^*=157$; roots $\frac{-1\pm\sqrt{157}}{2}$

157

x^3 + x^2 - 52*x + 64

comment: $4p=L^2+27M^2$ with $L=-14$, $M=4$

157

x^4 + x^3 + 20*x^2 - 206*x + 517

157

x^6 + x^5 - 65*x^4 + 160*x^3 + 20*x^2 - 208*x + 64

157

12:

x^12 + x^11 + 7*x^10 + 63*x^9 - 312*x^8 - 701*x^7 + 7047*x^6 - 33689*x^5 + 64030*x^4 - 41071*x^3 + 14685*x^2 + 86965*x + 53381

163

x^2 + x + 41

comment: $p^*=-163$; roots $\frac{-1\pm\sqrt{-163}}{2}$

163

x^3 + x^2 - 54*x - 169

comment: $4p=L^2+27M^2$ with $L=25$, $M=1$

163

x^6 + x^5 + 14*x^4 - 178*x^3 - 552*x^2 + 1854*x + 5023

163

x^9 + x^8 - 72*x^7 - 73*x^6 + 1482*x^5 + 1034*x^4 - 9637*x^3 - 1173*x^2 + 10087*x + 853

167

x^2 + x + 42

comment: $p^*=-167$; roots $\frac{-1\pm\sqrt{-167}}{2}$

173

x^2 + x - 43

comment: $p^*=173$; roots $\frac{-1\pm\sqrt{173}}{2}$

173

x^4 + x^3 + 22*x^2 + 292*x + 667

179

x^2 + x + 45

comment: $p^*=-179$; roots $\frac{-1\pm\sqrt{-179}}{2}$

181

x^2 + x - 45

comment: $p^*=181$; roots $\frac{-1\pm\sqrt{181}}{2}$

181

x^3 + x^2 - 60*x - 67

comment: $4p=L^2+27M^2$ with $L=7$, $M=5$

181

x^4 + x^3 + 23*x^2 + 215*x + 975

181

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

181

x^6 + x^5 - 75*x^4 + 104*x^3 + 918*x^2 - 2509*x + 1685

181

x^9 + x^8 - 80*x^7 + 53*x^6 + 1668*x^5 - 3314*x^4 - 4261*x^3 + 10795*x^2 - 2933*x - 1949

181

10:

x^10 + x^9 - 81*x^8 - 94*x^7 + 2418*x^6 + 3121*x^5 - 31973*x^4 - 43245*x^3 + 170860*x^2 + 209252*x - 201337

181

12:

x^12 + x^11 + 8*x^10 - 38*x^9 + 830*x^8 - 2182*x^7 + 8320*x^6 + 5533*x^5 - 11441*x^4 + 7976*x^3 + 385657*x^2 - 675065*x + 358525

191

x^2 + x + 48

comment: $p^*=-191$; roots $\frac{-1\pm\sqrt{-191}}{2}$

191

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

191

10:

x^10 + x^9 + 10*x^8 - 252*x^7 - 216*x^6 + 3244*x^5 + 17715*x^4 + 24287*x^3 + 16260*x^2 + 5200*x + 625

193

x^2 + x - 48

comment: $p^*=193$; roots $\frac{-1\pm\sqrt{193}}{2}$

193

x^3 + x^2 - 64*x + 143

comment: $4p=L^2+27M^2$ with $L=-23$, $M=3$

193

x^4 + x^3 - 72*x^2 - 205*x - 49

193

x^6 + x^5 - 80*x^4 - 125*x^3 + 1456*x^2 + 1744*x - 5184

193

x^8 + x^7 - 84*x^6 - 21*x^5 + 1981*x^4 + 63*x^3 - 14652*x^2 - 799*x + 30961

193

12:

x^12 + x^11 - 88*x^10 - 3*x^9 + 2617*x^8 - 1105*x^7 - 31612*x^6 + 16656*x^5 + 136836*x^4 - 25812*x^3 - 144990*x^2 - 40095*x + 6561

197

x^2 + x - 49

comment: $p^*=197$; roots $\frac{-1\pm\sqrt{197}}{2}$

197

x^4 + x^3 + 25*x^2 + 37*x + 1369

197

x^7 + x^6 - 84*x^5 - 217*x^4 + 1348*x^3 + 3988*x^2 - 1433*x - 1163

199

x^2 + x + 50

comment: $p^*=-199$; roots $\frac{-1\pm\sqrt{-199}}{2}$

199

x^3 + x^2 - 66*x + 59

comment: $4p=L^2+27M^2$ with $L=-11$, $M=5$

199

x^6 + x^5 + 17*x^4 + 247*x^3 + 1001*x^2 + 1871*x + 14485

199

x^9 + x^8 - 88*x^7 - 325*x^6 + 775*x^5 + 3447*x^4 - 1602*x^3 - 7354*x^2 - 3333*x + 121

199

11:

x^11 + x^10 - 90*x^9 - 115*x^8 + 2349*x^7 + 943*x^6 - 26327*x^5 + 21284*x^4 + 102168*x^3 - 217794*x^2 + 148930*x - 30647

Definition

The Gaussian period polynomial $\Psi_{p,k}(x)=\prod_{j=0}^{k-1}(x-\eta_j)$ of a prime $p$ and a divisor $k$ of $p-1$, where $\eta_j=\sum_{i=0}^{f-1}\zeta_p^{\,g^{j+ki}}$, $0\leq j<k$, are the $k$ Gaussian periods [4] of length $f=(p-1)/k$, with $g$ a primitive root modulo $p$ and $\zeta_p=e^{2\pi i/p}$.

Parameters

$p$

— prime ($p$ prime, $p\geq 5$)

$k$

— number of periods, the degree ($k\mid p-1$, $1<k<p-1$)

Formulas

(1)

$\sum_{j=0}^{k-1}\eta_j=\sum_{a=1}^{p-1}\zeta_p^{\,a}=-1$, so $\Psi_{p,k}(x)=x^k+x^{k-1}+\cdots$; and since every $\eta_j\equiv f\pmod{1-\zeta_p}$, $\Psi_{p,k}(x)\equiv(x-f)^k\pmod p$.

(2)

$\Psi_{p,2}(x)=x^2+x+\frac{1-p^*}{4}$ with $p^*=(-1)^{(p-1)/2}p$, that is, $\eta_0-\eta_1=\pm\sqrt{p^*}$ (Gauss) [4].

(3)

For $p\equiv 1\pmod 3$, writing $4p=L^2+27M^2$ with $L\equiv 1\pmod 3$, which determines $L$ and $M^2$: $\Psi_{p,3}(x)=x^3+x^2-\frac{p-1}{3}x-\frac{p(L+3)-1}{27}$ (Gauss [1], art. 358). The constant terms are the negatives of OEIS A394567 [5], the products of the three cubic periods.

(4)

For $f=2$, that is $k=(p-1)/2$, the periods are $\zeta_p^{\,a}+\zeta_p^{-a}=2\cos(2\pi a/p)$ and $\Psi_{p,(p-1)/2}(x)=\prod_{a=1}^{(p-1)/2}\bigl(x-2\cos(2\pi a/p)\bigr)$, the minimal polynomial of $2\cos(2\pi/p)$. The cosines are in $\cos(\pi x)$ for rational $x$ at $x=2a/p$.

(5)

For a Dirichlet character $\chi$ modulo $p$ whose order $d>1$ divides $k$, the Gauss sum $\tau(\chi)=\sum_{a=1}^{p-1}\chi(a)\zeta_p^{\,a}$ is $\sum_{j=0}^{k-1}\chi(g)^j\eta_j$, and conversely $\eta_j=\frac1k\sum_{\chi^k=1}\overline{\chi(g)}^{\,j}\tau(\chi)$ with $\tau(\chi_0)=-1$ for the trivial character [4]. With $g$ the least primitive root, the character with $\chi(g)=e^{2\pi i/d}$ has Conrey label $(p,n)$, $n\equiv g^{(p-1)/d}\pmod p$.

(6)

$\operatorname{disc}\Psi_{p,k}=(-1)^{r_2}p^{k-1}m^2$, where $(-1)^{r_2}p^{k-1}$ is the discriminant of $K_k$, with $r_2=k/2$ when $f$ is odd and $r_2=0$ when $f$ is even, and $m$ is the index of $\mathbb{Z}[\eta_0]$ in the ring of integers of $K_k$. For $f=2$, $m=1$ and $\operatorname{disc}\Psi_{p,(p-1)/2}=p^{(p-3)/2}$ [3], OEIS A203411 [6].

(7)

$\Psi_{p,1}(x)=x+1$ and $\Psi_{p,p-1}(x)=\prod_{a=1}^{p-1}(x-\zeta_p^{\,a})=\Phi_p(x)=x^{p-1}+\cdots+x+1$, the two rows not listed.

Comments

(8)

$\Psi_{p,k}$ depends on neither $g$ nor the numbering of the periods: another primitive root permutes the $\eta_j$. It is monic with integer coefficients and irreducible over $\mathbb{Q}$ of degree $k$, and its roots generate the unique subfield $K_k$ of degree $k$ of the cyclotomic field $\mathbb{Q}(\zeta_p)$; each $\eta_j$ is the trace of $\zeta_p^{\,g^j}$ from $\mathbb{Q}(\zeta_p)$ to $K_k$ [2]. The roots are real when $f$ is even and none of them is real when $f$ is odd, because $-1$ is a $k$-th power modulo $p$ exactly when $f$ is even. The name $\Psi_{p,k}$ is this table's; Gauss [1] and the literature after him call it the period equation, or period polynomial, of $k$ terms for $p$.

(9)

Two divisors of $p-1$ are left out. $k=1$ gives the single period $\eta_0=-1$ and $\Psi_{p,1}=x+1$; $k=p-1$ gives the periods $\zeta_p^{\,a}$ themselves, the roots of unity, and $\Psi_{p,p-1}=\Phi_p$, the cyclotomic polynomial, by (7).

(10)

For $k=2$ the two periods are the sums over the quadratic residues and the non-residues, and $\Psi_{p,2}=x^2+x+\frac{1-p^*}{4}$ with $p^*=(-1)^{(p-1)/2}p$ by (2), so its roots $\frac{-1\pm\sqrt{p^*}}{2}$ generate $\mathbb{Q}(\sqrt{p^*})$. For $p\leq 19$ those roots are entries of Algebraic numbers of degree 2, and the entry comments link them.

(11)

For $p=17$ the three rows $k=2,4,8$ are the equations of Gauss's construction of the regular $17$-gon [1]: the periods of length $8$, $4$ and $2$ are reached from $\mathbb{Q}$ by three successive quadratic extensions, and $2\cos(2\pi/17)=\zeta_{17}+\zeta_{17}^{16}$ is a root of $\Psi_{17,8}$.

(12)

An entry's comment gives, for $k=2$, the discriminant $p^*$ and the roots; for $k=3$, the $L$ and $M$ with $4p=L^2+27M^2$ that Gauss's formula (3) uses; for $f=2$, the roots as $2\cos(2\pi a/p)$; and for $p=17$ the construction of the $17$-gon. PARI's polsubcyclo(p, k), which returns a polynomial defining $K_k$, returns $\Psi_{p,k}$ itself on every entry here.

Programs

(P1)

PARI/GP

polsubcyclo(31, 5)      \\ x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

(P2)

Sage

p, k = 31, 5
f = (p - 1) // k; g = primitive_root(p)
K.<z> = CyclotomicField(p); R.<x> = ZZ[]
R(prod(x - sum(z^(g^(j + k*i) % p) for i in range(f)) for j in range(k)))   # x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5

References

[1]

C. F. Gauss, Disquisitiones Arithmeticae, 1801, Section VII, arts. 343–358; English translation by A. A. Clarke, Yale University Press, 1966.

[2]

H. Davenport, Multiplicative Number Theory, third edition, Graduate Texts in Mathematics 74, Springer, 2000, Chapter 3.

[3]

J. Brillhart, Note on the discriminant of certain cyclotomic period polynomials, Pacific Journal of Mathematics 152 (1992), no. 1, 15–19.

Links

[4]

Wikipedia: Gaussian period

[5]

OEIS A394567: product of the three cubic Gaussian periods for primes $p\equiv 1\pmod 3$, the negative of the constant term of $\Psi_{p,3}$

[6]

OEIS A203411: discriminant $p^{(p-3)/2}$ of $\Psi_{p,(p-1)/2}$, the period polynomial with $f=2$

Similar tables

Cyclotomic polynomials — $\Phi_p$ is the row $k=p-1$, not listed here

Algebraic numbers of degree 2 — holds the roots $\frac{-1\pm\sqrt{p^*}}{2}$ of $\Psi_{p,2}$ for $p\leq 19$

$\cos(\pi x)$ for rational $x$ — for $f=2$ the roots are $2\cos(2\pi a/p)$, twice its entries at $x=2a/p$

Gauss sums of primitive Dirichlet characters — the Gauss sums modulo $p$ and the periods are each other's Fourier transforms, (5)

Roots of unity — the $\zeta_p^{\,a}$ that are summed into the periods

Hilbert class polynomials $H_\Delta$ — its roots generate the ring class field of $\mathbb{Q}(\sqrt{\Delta})$, an abelian extension of an imaginary quadratic field, as the roots of $\Psi_{p,k}$ generate abelian extensions of $\mathbb{Q}$

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (every prime $p<200$ and every divisor $k$ of $p-1$ with $2\leq k\leq 12$ and $k<p-1$ is here, 158 entries)

How they were obtained:

Each polynomial is the product $\prod(x-\eta_j)$ computed exactly in Sage's cyclotomic field $\mathbb{Q}(\zeta_p)$, with every coefficient required to be a rational integer.

The generator requires it to equal PARI's polsubcyclo(p, k), a different algorithm, and requires the same product computed in ComplexBallField at 256 bits to enclose every coefficient in a ball of radius below $1/2$ (the worst radius over the table is $1.7\cdot 10^{-66}$), which determines an integer coefficient on its own. Before an entry is returned it must be monic, irreducible of degree $k$, with $x^{k-1}$ coefficient $1$; congruent to $(x-f)^k$ modulo $p$; equal to the closed forms of (2) and (3) when $k=2$ or $3$; have discriminant $(-1)^{r_2}p^{k-1}$ times a perfect square, and $p^{k-1}$ exactly when $f=2$; and have no root ball meeting the real axis when $f$ is odd. Outside the generator, the 21 cubic constant terms agree with OEIS A394567, the seven $f=2$ discriminants with OEIS A203411, the field each $\Psi_{p,k}$ defines is isomorphic to Sage's subfield of degree $k$ of $\mathbb{Q}(\zeta_p)$ for $p\leq 61$ (52 rows), the relation (5) holds exactly against Sage's Gauss sums for every character of order dividing $k$ with $p\leq 61$ (91 pairs) and in balls against the stored digits of the table of Gauss sums for $p\leq 50$ (71 pairs), the $f=2$ rows have the roots $2\cos(2\pi a/p)$ in balls, and $\Psi_{7,6}$, computed as a control and not stored, is $\Phi_7$. Two controls that must fail did: $\Psi_{7,3}$ is not polsubcyclo(7, 2), and $\Psi_{13,3}$ is not $(x-3)^3$ modulo $13$.