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A property of polynomials over a finite field
Published online by Cambridge University Press: 26 February 2010
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Let q be a power of an odd prime, [q] denote the Galois field GF(q) and write X(x) = xq − x. Let f(x) be a polynomial, having no linear factors, over [q], of positive degree, and write . Consider the continued fraction expansions
and
where the Ai(x) and aj,(x) are polynomials over [q] of degree ≥ 1 (if i ≥ 1, j ≥ 1). Plainly A0(x) = ao(x). Suppose that n = nf is the integer denned uniquely as the largest m such that
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