Let for a prime . We are interested in the structure of the ring . Via the Chinese Remainder Theorem, this question is equivalent to finding a factorization of over . Suppose factors in as
where each is an irreducible polynomial. What are the degrees of ? What is ?
We claim that where is the multiplicative order of mod .
This structure has been used in a beautifully instructive paper by Evra, Kowalski, and Lubotzky. (Yes, the one in Lubotzky-Philip-Sarnak.) In this paper, they have established connections among the Fourier transform, uncertainty principle, and dimension of ideals generated by polynomials in the ring above. We’ll talk about these connections in another post. The content of this post is Proposition 2.6(3) from their paper.