Exact values of Γ∗(10,p)

Abstract

For [Formula: see text] and [Formula: see text] a prime number, define [Formula: see text] to be the smallest positive integer [Formula: see text] such that any diagonal form [Formula: see text], with integer coefficients, has nontrivial zero over [Formula: see text] whenever [Formula: see text]. A special case of a conjecture attributed to Artin states that [Formula: see text]. It is well known that the equality occurs when [Formula: see text]. In this paper, we obtain the exact values of [Formula: see text] for all primes [Formula: see text] and, except for [Formula: see text], these values are much lower than those established in the conjecture, as might be expected.