A Generalization of a Lemma of Sullivan

Abstract

Consider an irreducible polynomial of the form f(X) = X ^p  − aX − b ∈ 𝔽[X] and α a root of f(X), where 𝔽 is a field of characteristic p. In 1975, F.J. Sullivan stated a lemma that provides the trace, taken with respect to the extension 𝔽(α)/𝔽, of elements of the form α ⁿ , where 0 ≤ n ≤ p ² − 1. We present a generalization of Sullivan's Lemma and provide another proof of the original lemma. We explain how computing Tr(α ⁿ ) for n < p ^r can be reduced to computing the traces Tr(α ^m ) for all m ≤ r(p − 1).

Key Words:

• Trace
• Trinomials

2000 Mathematics Subject Classification:

• 12E10
• 12F10

Notes

Communicated by T. Albu.