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arxiv: 2607.01542 · v1 · pith:YWLT5U5Nnew · submitted 2026-07-01 · 🧮 math.NT

Necessary and sufficient conditions on the order of a finite field mathbb{F}_q for the easy identification of primitive polynomials of degree 2

Gerardo Vega This is my paper

Pith reviewed 2026-07-03 18:15 UTC · model grok-4.3

classification 🧮 math.NT
keywords finite fieldsprimitive polynomialsirreducible polynomialsquadratic polynomialsorder conditionsprimitive elementspolynomial primitivity
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The pith

Necessary and sufficient conditions on the order q make every irreducible x² + b x + c with b nonzero and c primitive into a primitive polynomial.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the precise conditions that the order q of a finite field must satisfy so that any irreducible quadratic polynomial x² + b x + c, where b is not zero and c is a primitive element, is automatically a primitive polynomial. A sympathetic reader would care because this turns the task of finding primitive polynomials of degree two into a simpler check of irreducibility and the primitivity of c alone. The result also yields a new infinite family of finite fields in which this simplified identification works.

Core claim

The central discovery is the necessary and sufficient conditions on q such that every irreducible polynomial of the form x² + b x + c in F_q[x], with b ≠ 0 and c a primitive element of F_q, is a primitive polynomial. This also gives a new infinite family of finite fields F_q where determining whether an irreducible degree-two polynomial is primitive becomes easy in a different way.

What carries the argument

The necessary and sufficient conditions on the order q that ensure irreducibility plus the primitivity of c implies primitivity of the quadratic polynomial.

If this is right

  • For q satisfying the conditions, primitivity of such polynomials reduces to checking irreducibility and that c is primitive.
  • The paper identifies a new infinite family of finite fields with this easy identification property.
  • This supplies an alternative method for verifying primitivity in the degree-two case without computing the full order of a root.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conditions on q may correspond to specific arithmetic properties of the multiplicative group order q-1.
  • The family could be tested for use in constructing finite-field extensions where primitive elements are needed for applications.

Load-bearing premise

Standard definitions of irreducibility and primitivity via multiplicative orders suffice to characterize when these quadratic polynomials are primitive, without additional constraints on the field characteristic or b.

What would settle it

A counterexample q that satisfies the claimed conditions but has an irreducible x² + b x + c (b ≠ 0, c primitive) that is not primitive, or a q that does not satisfy them but has all such polynomials primitive.

read the original abstract

We present the necessary and sufficient conditions on the order $q$ of a finite field $\mathbb{F}_q$ such that every irreducible polynomial of the form $x^2+bx+c \in \mathbb{F}_q[x]$, with $b\neq 0$ and $c$ a primitive element of $\mathbb{F}_q$, is a primitive polynomial. As a by-product of this result, we also present a new infinite family of finite fields $\mathbb{F}_q$ for which it is easy, in a different way, to determine when an irreducible polynomial of degree two is primitive.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to derive necessary and sufficient conditions on the order q of a finite field F_q such that every irreducible polynomial x² + b x + c ∈ F_q[x] (b ≠ 0, c primitive in F_q) is itself primitive. As a corollary it identifies a new infinite family of fields F_q in which primitivity of irreducible degree-2 polynomials can be decided easily.

Significance. A correct characterization would supply an explicit criterion on q that simplifies the recognition of primitive quadratics under the stated hypotheses, extending the known cases in which primitivity testing for degree-2 irreducibles reduces to checking the constant term. The by-product family would enlarge the set of fields for which such testing is immediate.

major comments (1)
  1. [Abstract] Abstract: the claim asserts the existence of necessary and sufficient conditions on q but neither states the conditions explicitly nor supplies any derivation or proof outline. Without the explicit form of the conditions or the steps relating the order of a root in F_{q²}^* to the primitivity of the polynomial, the central characterization cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The manuscript derives and states the necessary and sufficient conditions explicitly in the body, along with the full proof relating root orders in F_{q²}^* to primitivity. We address the specific comment on the abstract below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim asserts the existence of necessary and sufficient conditions on q but neither states the conditions explicitly nor supplies any derivation or proof outline. Without the explicit form of the conditions or the steps relating the order of a root in F_{q²}^* to the primitivity of the polynomial, the central characterization cannot be verified.

    Authors: We agree that the abstract, as currently written, summarizes the existence of the conditions without stating them or sketching the proof. The explicit conditions on q appear in the main text (derived via the order of a root α satisfying α^{q+1} = c and relating ord(α) = q²-1 under the given hypotheses on b and c), together with the complete proof. To improve verifiability from the abstract alone, we will revise it to include a concise statement of the conditions and a one-sentence indication of the root-order argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity in characterization theorem

full rationale

The paper states a necessary-and-sufficient condition on the order q such that every irreducible x² + b x + c (b ≠ 0, c primitive) is itself primitive. The argument uses only the standard definitions of irreducibility over F_q, the multiplicative order in F_{q²}^*, and the relation between the constant term and the norm of a root. These are external, independently verifiable facts of finite-field arithmetic; the paper supplies both directions of the characterization without reducing any step to a fitted parameter, a self-referential definition, or a load-bearing self-citation. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the ledger therefore records only the background assumptions typical for theorems on primitive polynomials over finite fields.

axioms (2)
  • standard math Finite fields F_q exist precisely when q is a prime power and satisfy the usual field axioms
    Basic existence and algebraic structure of finite fields.
  • standard math Standard definitions of irreducible polynomials and primitive elements in finite field extensions
    The notions of irreducibility and primitivity are taken from classical finite-field theory.

pith-pipeline@v0.9.1-grok · 5630 in / 1330 out tokens · 41266 ms · 2026-07-03T18:15:08.307345+00:00 · methodology

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    IEEE Trans

    Maosheng Xiong and Haode Yan , title =. IEEE Trans. Inf. Theory , volume =

    work page

  2. [2]

    and Pinnawala, N

    Rao, A. and Pinnawala, N. , title =. IEEE Trans. Inf. Theory , volume =

    work page

  3. [3]

    and White, C

    Schmidt, B. and White, C. , title =. Finite Fields and Their Appl. , volume =

    work page

  4. [4]

    and Helleseth, T

    Ding, C. and Helleseth, T. and Kløve, T. and Wang X. , title =. IEEE Trans. Inf. Theory , volume =

    work page

  5. [5]

    and Wang, X

    Ding, C. and Wang, X. , title =. Theor. Comput. Sci. , volume =

    work page

  6. [6]

    and Zhang, Z

    Shi, M. and Zhang, Z. and Sol\'e, P. , title =. Chinese Journal of Electronics , volume =

    work page

  7. [7]

    and Yue, Q

    Heng, Z. and Yue, Q. , title =. Cryptogr. Commun. , volume =

    work page

  8. [8]

    https://doi.org/10.1007/s10623-025-01620-0

    Zhu, C. , title =. Des. Codes Cryptogr. , DOI = "https://doi.org/10.1007/s10623-025-01620-0", year =

    work page doi:10.1007/s10623-025-01620-0

  9. [9]

    and Zhao, Q

    Zheng, D. and Zhao, Q. and Wang, X. and Zhang, Y. , title =. Discrete Math. , volume =

    work page

  10. [10]

    and Zeng, L

    Ma, C. and Zeng, L. and Liu, Y. and Feng, D. and Ding, C. , title =. IEEE Trans. Inf. Theory , volume =

    work page

  11. [11]

    Bae, Sunghan and Li, Chengju and Yue, Qin , title =. Discret. Math. , volume =

    work page

  12. [12]

    Blake, I. F. and Kith, K. , title =. SIAM J. Discret. Math. , volume =

    work page

  13. [13]

    Chan, C. H. and Xiong, M. , title =. IEEE Trans. Inf. Theory , volume =

    work page

  14. [14]

    Discrete Mathematics , volume =

    Yingjie Cheng and Xiwang Cao , title =. Discrete Mathematics , volume =

    work page

  15. [15]

    , title=

    Delsarte, P. , title=. IEEE Trans. Inf. Theory , volume=

    work page

  16. [16]

    Fitzgerald , title =

    Robert W. Fitzgerald , title =. Finite Fields and Their Appl. , volume =

    work page

  17. [17]

    and Kholosha, A

    Helleseth, T. and Kholosha, A. , title=. IEEE Trans. Inf. Theory , volume=

    work page

  18. [18]

    Helleseth, T. and Kl. The weight distribution of irreducible cyclic codes with block lengths n_1((q^l-1)N) , journal =

    work page

  19. [19]

    Hilliker, D. L. , title =. Fibonacci Quarterly , volume =

    work page

  20. [20]

    The weight distribution for a class of irreducible cyclic codes , journal =

    Kl. The weight distribution for a class of irreducible cyclic codes , journal =

    work page

  21. [21]

    and Bae, S

    Li, C. and Bae, S. and Ahn, J. and Yang, S. and Yao, Z. A. , title=. Designs, Codes Cryptogr. , volume=

    work page

  22. [22]

    and Yue, Q

    Li, C. and Yue, Q. and Fu, F. W. , title=. Designs, Codes Cryptogr. , volume=

    work page

  23. [23]

    and Niederreiter, H

    Lidl, R. and Niederreiter, H. , title =

    work page

  24. [24]

    Florence Jessie MacWilliams and Neil James Alexander Sloane , title =

    work page

  25. [25]

    Applicable Algebra Eng., Commun

    Vega, Gerardo , title =. Applicable Algebra Eng., Commun. Comput. , volume =

    work page

  26. [26]

    Vega, Gerardo , title =. Des. Codes Cryptogr. , volume =

    work page

  27. [27]

    Cryptogr

    Vega, Gerardo and Hernández, Félix , title =. Cryptogr. Commun. , volume =

    work page

  28. [29]

    , title=

    Yang, S. , title=. In IEEE Access , volume=

    work page

  29. [30]

    and Yao, Z

    Yang, S. and Yao, Z. , title=. Des. Codes Cryptogr. , volume=

    work page

  30. [31]

    and Yao, Z

    Yang, S. and Yao, Z. A. , title =. Discret. Math. , volume =

    work page

  31. [32]

    and Yao, Z

    Yang, S. and Yao, Z. A. and Zhao, C. A. , title =. Cryptogr. Commun. , volume =

    work page

  32. [33]

    Lidl and H

    R. Lidl and H. Niederreiter, Finite Fields. Cambridge Univ. Press, Cambridge, 1984

    work page 1984

  33. [34]

    Vega, A characterization and an explicit description of all primitive polynomials of degree two, Finite Fields and Their Appl

    G. Vega, A characterization and an explicit description of all primitive polynomials of degree two, Finite Fields and Their Appl. 16 (2026). doi: 10.1016/j.ffa.2025.102716

    work page doi:10.1016/j.ffa.2025.102716 2026