Set of primes $p$ which $x^4-x^3-2x^2-2x-1$ completely factors in finite field of order $p$
Solution 1:
added. Caution: it is not a single quadratic form that represents the primes you want, it is two quadratic forms. (Monday) Indeed, we can take the two forms to be $$ x^2 + 95 y^2 \; , \; \; \; 5 x^2 + 19 y^2 \; , $$ as these represent the same odd numbers as $x^2 + xy + 24 y^2$ and $5 x^2 + 5 xy + 6 y^2;$ for either of these latter forms to be odd, we need $x(x+y)$ odd, therefore $x$ must be odd and $y$ must be even, leading to $y=2t$ and forms $(x+t)^2 + 95 t^2$ and $5(x+t)^2 + 19 t^2 \; .$
ORIGINAL:Quite surprised how this worked out. There is a 1973 article by Estes and Pall that proves that, for binary forms, the spinor kernel is the fourth powers in the form class group. At the end I have put a list up to 2000 of the relevant primes; 5 and 19 are not there, this is a simple program that just counts distinct roots mod p.
Your form discriminant is $-95,$ positive binary forms.
The forms in the principal genus that are not fourth powers are the pair of "opposites" $\langle 4,1,6 \rangle$ and $\langle 4,-1,6 \rangle.$ These represent the same primes, for which your polynomial splits as two irreducible quadratics. Added: if you prefer, you may use $\langle 9,4,11 \rangle,$ or $9x^2 + 4xy + 11 y^2,$ as it represents a subset of the same numbers, exactly the same odd numbers, and the same primes. Let's see: while it is necessary to allow $xy$ both positive and negative in searching for values of $9x^2 + 4xy + 11 y^2,$ we still get bounds on $|x|,|y|$ since $9x^2 + 4xy + 11 y^2 \geq \frac{95}{11} x^2$ and $9x^2 + 4xy + 11 y^2 \geq \frac{95}{9} y^2.$ The first few such primes are
11, 61, 101, 139, 149, 229, 271, 311, 359, 479,
499, 541, 571, 619, 631, 691, 701, 719, 761, 769,
881, 1031, 1049, 1061, 1069, 1259, 1279, 1301, 1489, 1499,
1669, 1721, 1759, 1811, 1831, 1871, 1949, 1999, 2069, 2099,
2221, 2239, 2251, 2381, 2441, 2531, 2671, 2851, 2969, 2999,
3049, 3079, 3089, 3121, 3209, 3331, 3361, 3389, 3659, 3691,
3779, 3881, 3911, 4001, 4051, 4111, 4159, 4229, 4241, 4339,
4409, 4481, 4561, 4621, 4721, 4729, 4751, 4759, 4871, 5021,
5039, 5051, 5059, 5099, 5261, 5419, 5441, 5519, 5591, 5641,
5659, 5669, 5701, 5711, 5801, 5839, 5849, 5869, 5939, 6011,
6029, 6199, 6271, 6389, 6469, 6571, 6581, 6599, 6619, 6689,
6781, 6841, 6961, 6971, 7079, 7129, 7229, 7321, 7331, 7351,
7459, 7549, 7639, 7649, 7829, 7901, 8101, 8111, 8209, 8219,
8231, 8269, 8291, 8329, 8369, 8521, 8669, 8689, 8741, 8941,
8969, 9041, 9049, 9091, 9181, 9221, 9239, 9371, 9391, 9421,
9479, 9511, 9619, 9649, 9791, 9829, 9859, 10039, 10079, 10151,
10271, 10391, 10531, 10651, 10789, 10891, 10979,
The principal form is $\langle 1,1,24 \rangle.$ With these primes, your polynomial splits as four distinct linear factors. Added: if you prefer, you may use $\langle 1,0,95 \rangle,$ or $x^2 + 95 y^2,$ as it represents a subset of the same numbers, exactly the same odd numbers, and the same primes.
131, 239, 389, 419, 461, 821, 859, 919, 1051, 1109,
1531, 1601, 1879, 1901, 2011, 2399, 2411, 2609, 2699, 2791,
2971, 3011, 3041, 3469, 3541, 3559, 3671, 3709, 4139, 4219,
4261, 4349, 4451, 4679, 4691, 4789, 4799, 4951, 5101, 5231,
5279, 5479, 5821, 6089, 6229, 6521, 6959, 7151, 7559, 7699,
7759, 7949, 7951, 8081, 8179, 8461, 8599, 8681, 8719, 9011,
9029, 9311, 9319, 9349, 9431, 9631, 9661, 9811, 9839, 9941,
10169, 10181, 10399, 10459, 10499, 10589, 10739, 10831, 11059, 11321,
11701, 12071, 12101, 12641, 12791, 12829, 13171, 13259, 13399, 13469,
13649, 13681, 13729, 13799, 13841, 14029, 14411, 14419, 14779, 14869,
15091, 15361, 15439, 15739, 15881, 15889, 15971, 16061, 16091, 16189,
16231, 16319, 16631, 16649, 17021, 17239, 17299, 17351, 17401, 17519,
17579, 17581, 18061, 18149, 18169, 18251, 18401, 18701, 19009, 19139,
19301, 19609, 19709, 20261, 20411,
The other fourth power is $\langle 5,5,6 \rangle.$ This represents $5$ and $19,$ for which the polynomial has repeat roots. Other primes represented also split your polynomial into distinct linear factors. Added: if you prefer, you may use $\langle 5,0,19 \rangle,$ or $5x^2 + 19 y^2,$ as it represents a subset of the same numbers, exactly the same odd numbers, and the same primes.
5, 19, 191, 199, 251, 349, 491, 709, 739, 809,
929, 1151, 1201, 1289, 1429, 1451, 1559, 1619, 1621, 2039,
2129, 2281, 2341, 2551, 2591, 2741, 2819, 2861, 3019, 3329,
3391, 3539, 3581, 3769, 3919, 3931, 4091, 4129, 4519, 4831,
4861, 4889, 4909, 4919, 5179, 5381, 5431, 5521, 5749, 5861,
6091, 6211, 6659, 6661, 6761, 7001, 7039, 7069, 7369, 7411,
7529, 7541, 7681, 8171, 8699, 8779, 8821, 8839, 8861, 9241,
9281, 9539, 9601, 9739, 9851, 9929, 10321, 10429, 10771, 10799,
10949, 11069, 11119, 11329, 11549, 11789, 11971, 11981, 12119, 12281,
12451, 12671, 12689, 12841, 12889, 13001, 13249, 13309, 13339, 13499,
13691, 13781, 13931, 14159, 14221, 14551, 14561, 14731, 14741, 14831,
15131, 15149, 15401, 15511, 15679, 15749, 15809, 16699, 16729, 16879,
17011, 17231, 17599, 17789, 17791, 18059, 18089, 18289, 18379, 18439,
18541, 18859, 18959, 18979, 19289, 19391, 19501, 19861, 20071, 20149,
20201, 20341, 20479, 20759, 20771, 20879,
Here is the form class group for discriminant $-95$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./classGroup
Absolute value of discriminant?
95
Discr -95 = 5 * 19 class number 8
all
95: < 1, 1, 24> Square 95: < 1, 1, 24>
95: < 2, -1, 12> Square 95: < 4, -1, 6>
95: < 2, 1, 12> Square 95: < 4, 1, 6>
95: < 3, -1, 8> Square 95: < 4, -1, 6>
95: < 3, 1, 8> Square 95: < 4, 1, 6>
95: < 4, -1, 6> Square 95: < 5, 5, 6>
95: < 4, 1, 6> Square 95: < 5, 5, 6>
95: < 5, 5, 6> Square 95: < 1, 1, 24>
squares
95: < 1, 1, 24>
95: < 4, -1, 6>
95: < 4, 1, 6>
95: < 5, 5, 6>
fourths
95: < 1, 1, 24>
95: < 5, 5, 6>
Discriminant -95 h : 8 Squares : 4 Fourths : 2
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
=====================================
These are the first few primes (NOT $5,19$ because they have repeated roots) for which the polynomial has four distinct roots.
jagy@phobeusjunior:~$ ./count_roots
131 count 1
191 count 2
199 count 3
239 count 4
251 count 5
349 count 6
389 count 7
419 count 8
461 count 9
491 count 10
709 count 11
739 count 12
809 count 13
821 count 14
859 count 15
919 count 16
929 count 17
1051 count 18
1109 count 19
1151 count 20
1201 count 21
1289 count 22
1429 count 23
1451 count 24
1531 count 25
1559 count 26
1601 count 27
1619 count 28
1621 count 29
1879 count 30
1901 count 31
2011 count 32
Note that the version of the polynomial used on the field website is something like $ -x^4 \cdot f\left(\frac{-1}{x}\right)$
Added Monday lunchtime: perhaps a little more attractive to say the primes giving four linear factors are represented by the two forms $$ x^2 + 95 y^2 \; , \; \; \; 5 x^2 + 19 y^2 \; , $$ while the primes that give two irreducible quadratic are represented by $$ 9 x^2 \pm 4xy + 11 y^2 $$