Let q be a prime power, let F_q be the field with q elements, and let overlineF_q denote an algebraic closure of F_q. For f ∈ F_q[x] of degree d ≥ 1, put Phi_f(x,y) ;=; fracf(x)-f(y)x-y ;∈; F_q[x,y], which is a polynomial of total degree d-1. Call f exceptional over F_q if every irreducible factor of Phi_f in F_q[x,y] fails to be irreducible in overlineF_q[x,y]; that is, Phi_f has no absolutely irreducible factor over F_q. For every prime power q and every integer d ≥ 1 with gcd(d,, q-1) > 1, there is no f ∈ F_q[x] of degree d that is exceptional over F_q.

Carlitz–Wan conjecture

About 60 years old · traced to 1966

Let $q$ be a prime power, let $\mathbb{F}_q$ be the field with $q$ elements, and let $\overline{\mathbb{F}_q}$ denote an algebraic closure of $\mathbb{F}_q$ .

For $f \in \mathbb{F}_q[x]$ of degree $d \ge 1$ , put

\Phi_f(x,y) \;=\; \frac{f(x)-f(y)}{x-y} \;\in\; \mathbb{F}_q[x,y],

which is a polynomial of total degree $d-1$ . Call $f$ exceptional over $\mathbb{F}_q$ if every irreducible factor of $\Phi_f$ in $\mathbb{F}_q[x,y]$ fails to be irreducible in $\overline{\mathbb{F}_q}[x,y]$ ; that is, $\Phi_f$ has no absolutely irreducible factor over $\mathbb{F}_q$ .

For every prime power $q$ and every integer $d \ge 1$ with

\gcd(d,\, q-1) > 1,

there is no $f \in \mathbb{F}_q[x]$ of degree $d$ that is exceptional over $\mathbb{F}_q$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The Carlitz--Wan conjecture on degrees of exceptional polynomials
Let $q$ be a prime power, and let $f(X)\in\mathbb{F}_q[X]$ be an exceptional polynomial, meaning that $f$ induces a bijection on $\mathbb{F}_{q^k}$ for infinitely many positive integers $k$ . Carlitz--Wan conjecture. The degree of $f$ is coprime to $q-1$ . The conjecture generalizes the Carlitz conjecture by imposing an arithmetic restriction on the degree of every exceptional polynomial over a finite field. The source states that the conjecture in full generality was first proved by Lenstra in 1994.
source: Zhiguo Ding, Wei Xiong and Qifan Zhang, “Exceptional extensions of local fields and the Carlitz–Wan conjecture”, arXiv:2505.12877 (2025).

References

Primary source

Wikipedia

Additional references

Wikipedia, Carlitz–Wan conjecture, the article this problem comes from.

algebra unsolved-problem

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

The conjecture was proved decades ago, and a 2026 paper claims another proof, but that new argument has not been independently verified.

The Carlitz–Wan conjecture asserts that an exceptional polynomial of degree $d$ over $\mathbb{F}_q$ cannot exist when $\gcd(d,q-1)>1$ . Carlitz posed the odd-field, even-degree case in 1966; Wan proposed the general form in 1993.

Known results

Carlitz, 1966: posed the special case of odd $q$ and even $d$ .
Fried, Guralnick, and Saxl, 1993: proved that special case.
Lenstra, 1995: proved the general conjecture; later accounts attribute versions of the proof to his communications and to Cohen and Fried.
A 2025 paper records a stronger rational-function statement, implying the polynomial case.

2026 claimed new proof

Hu and Zhang’s 2026 arXiv paper claims a new proof using Weil’s conjecture and Bombieri–Katz point-counting estimates. Its revised version states the contradiction argument and concludes the theorem, but the retrieved sources contain no independent verification, error report, or retraction.

Current status (as of September 2026): The conjecture is regarded as settled by Lenstra’s proof, while Hu and Zhang’s 2026 alternative proof remains an unverified claim.

Sources

No solutions have been posted yet.

The Carlitz--Wan conjecture on degrees of exceptional polynomials

Primary source

Additional references

Progress summary