The size of algebraic integers with many real conjugates

In this paper we show that the relative normalised size with respect to a number field K of an algebraic integer alpha not equal -1, 0, 1 is greater than 1 provided that the number of real embeddings s of K satisfies s >= 0.828n, where n = [K : Q]. This can be compared with the previous much more restrictive estimate s >= n 0.192 root n/log n and shows that the minimum m(K) over the relative normalised size of nonzero algebraic integers alpha in such a field K is equal to 1 which is attained at alpha = +/- 1. Stronger than previous but apparently not optimal bound for m(K) is also obtained for the fields K satisfying 0.639 <= s/n < 0.827469 . . . . In the proof we use a lower bound for the Mahler measure of an algebraic number with many real conjugates