On the measure of a nonreciprocal algebraic number

Let M(alpha) be the Mahler measure of an algebraic number alpha and let G(a lpha) be the modulus of the product of logarithms of absolute values of its conjugates. We prove that if alpha is a nonreciprocal algebraic number of degree d greater than or equal to 2 then M(alpha)(2)G(alpha)(1)/d greater t han or equal to 1/2d. This estimate is sharp up to a constant. As a main to ol for the proof we develop an idea of Cassels on an estimate for the resul tant of alpha and 1/alpha. We give a number of immediate corollaries, e.g., some versions of Smyth's inequality for the Mahler measure of a nonrecipro cal algebraic integer from below.