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A generalization of a lemma of Sullivan. (English) Zbl 1271.12002

Let \({\mathbb F}\) be a field of characteristic \(p\). The trace map of powers of a root of an Artin–Schreier polynomial has been used, among other things, to find bounds of exponents of class groups in congruence function fields by M. Madan and D. Madden [Acta Arith. 32, 183–205 (1977; Zbl 0371.12010)].
Now consider \(f(X) = X^p - aX - b\), an irreducible trinomial with coefficients in the field \(\mathbb F\) and let \(\alpha\) be a root of \(f(X)\). F. J. Sullivan [Arch. Math. 26, 253–261 (1975; Zbl 0331.14016)] stated without proof a lemma that provides the trace, with respect to \(\mathbb F(\alpha)/\mathbb F\), of \(\alpha^n\) for \(0 \leq n \leq p^2 - 1\) in terms of \(a\) and \(b\).
The lemma has been used by several authors, including Sullivan, to obtain the entries of a matrix related to the Hasse-Witt map of some \(p\)-extensions of function fields of characteristic \(p\).
For the proof of the lemma, Sullivan referred to another paper of himself which was never published. M. Rzedowski-Calderón, P. Lam-Estrada and M. R. Maldonado-Ramírez [Aguilar, M. (ed.) et al., Memorias de la Sociedad Matemática Mexicana. Aportaciones Mat., Comun. 40, 133–142 (2009; Zbl 1220.11134)] presented a proof of the lemma. In the paper under review, the authors give a new proof of Sullivan’s Lemma and a generalization of it. The generalization is based on the observation that the computation of \(\roman{Tr}(\alpha^n)\) for \(n < p^r\) can be reduced to the computation of \(\roman {Tr}(\alpha^m)\) for all \(m < r(p -1)\). Some tools to achieve this reduction are Viète’s relations and Newton’s identities.

MSC:

12E10 Special polynomials in general fields
12F10 Separable extensions, Galois theory

Keywords:

trace; trinomials
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References:

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