Summary
This paper continues the author's excursions into the arithmetic of polynomials over finite fields. For monic polynomials A, B ε GF[q, x] where p is a prime, q=pdand d ⩾ 1: The divisor B of A is a bi-unitary divisor of A provided 1 is the greatest common unitary divisor of the polynomials B and A/B, and we say that A is bi-unitary perfect (b.u.p.) over GF(q) provided A equals the sum σ**(A) of the distinct bi-unitary divisors of A in GF[q, x]. A diversity of b.u.p. polynomials over GF(q) is found, some of which are neither perfect nor unitary perfect. For p > 2 we can only conjecture a characterisation of the b.u.p. polynomials which split in GF[p, x], so several open questions remain. Examples of non-splitting b.u.p. polynomials over GF(p) are given for p=2, 3, 5 which, in turn, allow the construction of such examples over GF(pd) for these p.
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Beard, J.T.B. Bi-unitary perfect polynomials overGF(g) . Annali di Matematica pura ed applicata 149, 61–68 (1987). https://doi.org/10.1007/BF01773925
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DOI: https://doi.org/10.1007/BF01773925