A complete classification of a primitive cubic family
Henry Zweiman
September 9, 2026
Abstract
We prove that a primitive polynomial over , with primitive in , exists if and only if the characteristic is not three. This determines all surviving cases of a conjecture of Awasthi and Sharma, whose case was recently disproved. A Kummer parametrization retains the coefficient conditions exactly and gives uniform character-sum estimates, with different main terms in odd and even characteristic. A sieve proves existence for every , and independently verified finite-field certificates cover the remaining 261 prime powers. The proof uses the incomplete-character-sum theorem of Fu and Wan and includes an explicit accounting of its geometric hypotheses.
1. Introduction and statement
A monic polynomial of degree over is primitive if its roots generate the multiplicative group . Awasthi and Sharma [1, Conjecture 9.1] asked whether there always exists a primitive polynomial
with primitive in . Their surrounding discussion takes prime. Sharma [3, Section 4] disproved the case and considered the broader prime-power setting. We give the complete classification in that setting.
Theorem 1.1.
Let be any prime power. There exists a primitive element for which (1) is primitive if and only if .
The positive direction is partly computer assisted: a uniform argument reduces it to a specified finite interval, and 261 explicit certificates cover that entire interval. The files witnesses.json and check.py provide the certificates and a verifier using only exact integer and polynomial arithmetic. The proof does not infer the infinite conclusion from a sample.
We also obtain a counting result. Write for the number of constants in Theorem 1.1, and put
where is the number of distinct prime divisors of .
Theorem 1.2.
If has odd characteristic different from three, then
If is even, then
In characteristic three, .
Thus the even-characteristic family has twice the leading density. Since , both estimates have a relative error tending to zero as tends to infinity in the respective regime.
The character estimates come from Fu and Wan [2, Proposition 2.1 and Theorem 4.2]. The Kummer parametrization below enforces both prescribed coefficients before applying those estimates. This is essential: multiplicative freeness in an extension field does not enforce membership in a smaller field. We discuss the distinction from the sufficient condition in [3] at the end.
2. The obstruction and a parametrization
Lemma 2.1.
In characteristic three, no polynomial (1) can be both irreducible and have primitive constant term.
Proof.
The discriminant of in characteristic three is . An irreducible cubic over an odd finite field has nonzero square discriminant: Frobenius permutes its roots as a three-cycle and therefore fixes their Vandermonde product. Since is an odd square, is a square in . Consequently must be a square. A generator of the even-order cyclic group is not a square.
From now on the characteristic differs from three. Then . Choose a noncube and a root of . This cubic is irreducible, since a reducible cubic has a root. Hence
The element is a primitive cube root of unity in , and the three conjugates of are .
For , define its subtrace by
Every has a unique expression with . Direct expansion of the three conjugates gives
Lemma 2.2.
If , the elements of having trace and subtrace are exactly
each occurring once. If , they form the union of the two lines
whose only common element is .
Proof.
For , equations (2) give and . Take and solve for . In characteristic two they give and . Uniqueness of the basis expansion proves both the parametrizations and the intersection assertion.
A primitive has degree three over . Its minimal polynomial is
Moreover, the norm of a generator of generates . Negation preserves generators of when is even or . In the latter case, if generates the group and , then remains coprime to : it remains odd and is unchanged modulo every odd prime divisor. Since , one of these two cases always applies.
It follows that every primitive element in Lemma 2.2 gives a polynomial counted by , and every such polynomial has exactly three counted roots. In characteristic two we may count the two lines with multiplicity at , since is not primitive in .
3. Incomplete character sums
We record the precise specialization of the external character-sum theorem that we use. Let , and let be an open subset of obtained by removing the zeros and poles of all three coefficient conjugates of . For a nontrivial multiplicative character of , the Kummer sheaf of has rank one, weight zero, and only tame ramification. Fu–Wan tensor induction produces a rank-one sheaf on whose trace at is . After base change, its three Kummer factors have character weights on the coefficient conjugates; see [2, Proposition 2.1].
If some zero belongs to exactly one conjugate and is simple, its inertia exponent in this tensor product is one of . These integers are coprime to the order of , which divides . The induced sheaf is therefore geometrically nonconstant. If the removed set has geometric points, [2, Theorem 4.2], with genus zero and zero Swan conductors, gives
The puncture count in (5) is the actual union of the three supports. This is sharper here than counting their degrees separately. These observations check the geometric nonconstancy hypothesis; the mere nontriviality of would not suffice.
Lemma 3.1.
For every nontrivial multiplicative character of , we have
Each has a Frobenius orbit of size three. The orbits could meet only if were a cube root of unity. That would imply , impossible for . Thus the three conjugate functions have six disjoint simple zeros. Their poles are and , both simple, and there are eight punctures in total. No zero lies in , so . Apply (5).
For either line in (4), its zero and the two coefficient-conjugate zeros are distinct and lie outside . The pole is . Hence there are four punctures and . The same nonconstancy check and (5) prove the second bound.
For a divisor , call -free if it is not an -th power for any prime . Let count -free values in the parameter multiset of Lemma 2.2, counting the common point of the two lines twice in characteristic two. Every value is nonzero. The standard indicator is
This follows by multiplying the prime-divisor indicators in the cyclic group .
Set
The trivial character contributes . Lemma 3.1 and (6) imply
Since , this proves Theorem 1.2, including the obstruction from Lemma 2.1.
4. A sieve and a finite cutoff
Lemma 4.1.
Write , where is squarefree and the are distinct primes not dividing . Suppose and
Then whenever
For , the sufficient condition is .
Proof.
Pointwise on any multiset of elements of , the -free indicators give
Indeed, an element which is not -free contributes zero on the right; a -free element missing at least one further prime condition contributes at most zero, and an -free element contributes one.
Subtracting the character expansions for and leaves the squarefree character orders divisible by . There are such orders, with normalized total weight one for each. Therefore
Let , and write for the ordinary primes. Take to contain the smallest prime divisors of . For , put
using only choices with . For , put . The actual is at least , so Lemma 4.1 applies if .
For , choose respectively
Exact rational calculation gives in every case. For , Table 1 gives and an integer satisfying
Since has distinct prime factors, the first inequality implies , and the second proves the desired sieve condition.
For the infinite tail, the exact base inequality is
The next prime is , as are all subsequent primes. Induction therefore proves for every . Hence , which implies . The direct case suffices.
All finite rational inequalities, prime products, and the tail base inequality are checked by the supplied verifier. This completes the cutoff proof.
Table 1. Exact finite sieve certificates for (11).
16
3
1724
30
3
6891
44
4
15968
17
3
1949
31
3
7469
45
4
16726
18
3
2195
32
3
8090
46
4
17515
19
3
2456
33
3
8752
47
4
18316
20
3
2736
34
3
9470
48
4
19129
21
3
3040
35
4
10129
49
4
19969
22
3
3363
36
4
10701
50
4
20841
23
3
3710
37
4
11290
51
4
21743
24
3
4078
38
4
11897
52
4
22671
25
3
4464
39
4
12525
53
4
23634
26
3
4878
40
4
13171
54
4
24617
27
3
5326
41
4
13837
55
4
25630
28
3
5807
42
4
14529
56
4
26673
29
3
6329
43
4
15235
5. Finite certificates and completion of the proof
For each of the 261 prime powers with , the file witnesses.json supplies a monic polynomial of degree over and an exponent . If denotes the residue class of , the certificate verifies that has exact order in . This proves that the quotient is a field: the cyclic subgroup generated by already contains all nonzero residue classes. It is therefore a specified model of .
The exponent satisfies , and . In the quotient by , the residue class of is checked to satisfy
Thus its order is exactly . Independently, the verifier checks
A cubic with no root in is irreducible, so this also verifies the field property of the cubic quotient directly. Consequently each certificate gives a primitive polynomial of the required form.
There is no unverified factorization in (12). The full prime support of is obtained by factoring the four integers in
For this finite interval, each factor is less than . Trial division is enough, and the verifier removes all resulting prime powers from and checks that the residual factor is one. It likewise verifies the complete set of prime powers below the cutoff, rather than merely reading a claimed count.
The witness generator uses C++ integer encodings of finite-field elements. The separate Python verifier uses coefficient tuples and independent polynomial arithmetic. It verifies the base-field generator, the primitive constant, cubic irreducibility, all root-order tests, the 261-case coverage, and every finite inequality in Proposition 4.2. Neither implementation uses floating-point arithmetic for a mathematical decision.
Lemma 2.1 excludes characteristic three. In all other characteristics, Proposition 4.2 proves existence when . The finite certificates prove existence for every remaining prime power . The norm argument after Lemma 2.2 ensures that all counted constants are primitive. These cases exhaust the theorem.
6. Relation to earlier claims and further questions
The case was already disproved in [3]; we do not claim priority for that counterexample. The new conclusion is the complete classification of the surviving cases, together with the two uniform counting estimates. The general degree- existence problem for polynomials , with nonconstant coefficients in a subfield, remains outside the theorem.
The sufficient-condition argument in [3, Section 5] counts pairs in which is -free in . That condition does not require to lie in , so it is not equivalent to the stated primitive-constant condition. Its general conclusion also conflicts with a norm obstruction: if is odd and , the norm of a primitive root would be , while a primitive has nonprimitive negative. Taking and arbitrarily large meets the characteristic hypothesis of its Theorem 5.1 and exhibits this incompatibility. Our proof for the cubic family uses no such equivalence.
The parametrization gives a more general route for prescribed-coefficient questions: impose the trace equations first, then check that the induced multiplicative characters are geometrically nonconstant on the resulting curve. The characteristic-three obstruction here shows that this last condition cannot be omitted. Extending the classification to other pairs of prescribed cubic coefficients, or to higher-degree subfield families, requires a new analysis of those curves and their exceptional characters.
References
[1] A. Awasthi and R. K. Sharma, Primitive transformation shift registers over finite fields, Journal of Algebra and Its Applications 18 (2019), 1950171. doi:10.1142/S0219498819501718. Author revision dated August 18, 2018; earlier version arXiv:1612.05367.
[2] L. Fu and D. Wan, A class of incomplete character sums, Quarterly Journal of Mathematics 65 (2014), 1195–1211. doi:10.1093/qmath/hau012. We use the corrected arXiv:1303.3650v3, January 6, 2025.
[3] A. K. Sharma, On the existence of primitive polynomials over finite fields, arXiv:2608.07262v1, August 7, 2026.