We prove that the local A 1 -degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local A 1 -degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers, and by Kass and Wickelgren's work on the Scheja-Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren relating the Scheja-Storch form and the local A 1 -degree.

No abstract

No abstract

No abstract

Given n ∈ N, we study the conditions under which a finite field of prime order q will have adjacent elements of multiplicative order n. In particular, we analyze the resultant of the cyclotomic polynomial Φ n (x) with Φ n (x + 1), and exhibit Lucas and Mersenne divisors of this quantity. For each n = 1, 2, 3, 6, we prove the existence of a prime q n for which there is an element α ∈ Z qn where α and α + 1 both have multiplicative order n. Additionally, we use algebraic norms to set analytic upper bounds on the size and quantity of these primes.