Abstract. We construct some locally unbounded topological fields having topologically nilpotent elements; this answers a question of Heine. The underlying fields are subfields of fields of formal power series. In particular, we get a locally unbounded topological field for which the set of topologically nilpotent elements is an open additive subgroup. We also exhibit a complete locally unbounded topological field which is a topological extension of the field of p-adic numbers; this topological field is a missing example in the classification of complete first countable fields given by Mutylin.
Let p be prime, q = p m , and q − 1 = 7s. We completely describe the permutation behavior of the binomial P(x) = x r (1 + x es ) (1 ≤ e ≤ 6) over a finite field F q in terms of the sequence {a n } defined by the recurrence relation a n = a n−1 + 2a n−2 − a n−3 (n ≥ 3) with initial values a 0 = 3, a 1 = 1, and a 2 = 5.
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