Class Number Parity for the pth Cyclotomic Field

We study the parity of the class number of the pth cyclotomic field for p prime. By analytic methods we derive a parity criterion in terms of polynomials over the field of 2 elements. The conjecture that the class number is odd for p a prime of the form 2q + 1 , with q prime, is proved in special cases, and a heuristic argument is given in favor of the conjecture. An implementation of the criterion on a computer shows that no small counterexamples to the conjecture exist.