Cyclic p-roots of prime lengths p and related complex Hadamard matrices

TL;DR

This paper proves that for every prime p, the set of cyclic p-roots in complex space is finite and provides an exact count, impacting the understanding of complex Hadamard matrices of prime size.

Contribution

It establishes the finiteness of cyclic p-roots for prime p and derives an explicit formula for their number, advancing the classification of complex circulant Hadamard matrices.

Findings

Number of cyclic p-roots is (2p-2)!/(p-1)!^2

Finite set of cyclic p-roots for prime p

Upper bound on complex circulant Hadamard matrices

Abstract

In this paper it is proved, that for every prime number p, the set of cyclic p-roots in C^p is finite. Moreover the number of cyclic p-roots counted with multiplicity is equal to (2p-2)!/(p-1)!^2. In particular, the number of complex circulant Hadamard matrices of size p, with diagonal entries equal to 1, is less or equal to (2p-2)!/(p-1)!^2.