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Introduction. There are a number of interesting results connecting the geometry of the roots of a polynomial with integer coefficients with algebraic properties of the roots. The best known is Kronecker’s famous result which states that if a polynomial with integer coefficients has all its zeros on the unit circle then these must all be roots of unity [1]. One cannot extend this too far, as the existence of Salem numbers shows: a Salem polynomial f(x) is a monic irreducible polynomial with integer coefficients which has one positive root outside the unit circle, one inside the circle and the remainder on the circle. The roots on the unit circle are obviously not roots of unity since f is irreducible. In fact, there are no nontrivial multiplicative relations between these roots, as shown by Pisot (see [3, p. 32]). Another related result, due to Smyth [4], is that if f(x) is a Pisot polynomial then it has no two conjugates of equal modulus, except for pairs of complex conjugate roots. A Pisot polynomial is a monic polynomial with integer coefficients with a single positive root outside the unit circle and the rest inside. A generalization of this due to Mignotte [2] is that there is no nontrivial multiplicative relation between the zeros of a Pisot polynomial. An obvious way to construct polynomials with roots of equal modulus is to take f(x) to be of the form g(x), so the roots of f are all of the mth roots of the roots of g. Of course, even if g is irreducible, f may or may not be irreducible. Michael Hollander asked the author for an example of an irreducible f(x) with a positive root α1 of largest modulus and m−1 other roots of modulus α1 but which is not of this form. The purpose of this note is to show that such examples do not exist.

Irreducible polynomials with many roots of maximal modulus