MATHEMATICS OF COMPUTATION VOLUME 51, NUMBER 184 OCTOBER 1988, PAGES 837-839 A Remark on a Paper by Wang: Another Surprising Property of 42 By J. A. Abbott and J. H. Davenport Abstract. We give a counterexample to a bound quoted in Wang's paper on polynomial factorization over algebraic number fields. We also give an alternative to that bound which seems not to have been published before. Currently the best algorithms for factorizing polynomials over algebraic number fields perform a factorization over a finite field, refine this factorization by Hensel lifting to a suitable prime power, and finally deduce the true factorization. It is thus necessary to estimate how far the modular factorization has to be lifted so we can be sure of deducing the true irreducible factors. This estimate comes from a bound on the "sizes" of coefficients of the factors. We introduce some notation: / is the polynomial to be factorized over the field Q(0), where 6 is an algebraic integer of degree m with minimal polynomial me{x) € Z[x]. Let \\0\\= max{\(t>($)\ : <¡>: Q(0) -» C} (noting that the product of the </>(0) is the trailing coefficient, so at least one \<p{6)\ is at least one). If R = Ylj=o cix^ l% a polynomial with complex coefficients, define ||P||2 = yJYl, Ie.? I2> following [2]. Recall Gauss's Lemma: let O be the ring of integers in Q(0); if f(x) € 0[x] factorizes in Q(0)[z] then the factors can be taken to lie in 0[x]. So we compute [1] the defect: the largest denominator, A, that occurs in O, i.e., O Ç ¿Z[ö]. In fact, an integer multiple of A will suffice [4]. We now know how big the denominator can be, so we estimate the numerator. We can regard f(x) as an element of C[x] and determine the maximum magnitude, B, of any coefficient of any factor in C[a;]. Thus our problem may be restated as: given that | Y^i=~0 ci^/^\ — B for all embeddings Q{0) —* C, find a bound on |c,-|. Let M be the Vandermonde matrix (\ 1 VI On jm-1 3m —1 so we can write Mc = b with b lying in a cube of side 2Aß centered on 0. In [3] Wang quotes Weinberger (presumably [4]) as having proved (1) Aßm!| |det(M)| Received August 27, 1987; revised January 25, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 12D10. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page 837 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use