Modular Periodicity of Binomial Coefficients

We prove that if the signed binomial coefficient (−1) ` k i ́ viewed modulo p is a periodic function of i with period h in the range 0 ≤ i ≤ k, then k + 1 is a power of p, provided h is prime to p and not too large compared to k. (In particular, 2h ≤ k suffices.) As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H < G, and such that 1− α ∈ G for all α ∈ G \H, then G ∪ {0} is a subfield.