By Henry B. Mann.

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A~J coincide. This however leads to a contradiction. 1. Distribution of Integers in the Complex Plane 51 and a E Gal(K/Q) carries a into aP, then we get, denoting by N the order of a, thus a either zero or a root of a unity, contrary to our assumption. D 5. 7. (Robinson [62]) If I is an interval on the real axis having length > 4, then one can find in I infinitely many full sets of conjugates of algebraic integers. Moreover, for the particular intervals [-2- t:,2 + t:] with a positive f one can find such sets not containing any number of the form 2cos(1rr) with r E Q.

26 shows that Pv = 1rRv with 1r E P\P 2 , thus PvC PRv. On the other hand, every x E PRv can be written as x = a1b 1+ · ·+akbk with ai E R, v(ai) 2: 1 and bi E K, v(bi) 2: 0. Hence v(x) 2: mini{v(aibi)} 2: 1. This implies x E Pv, hence PRv C Pv. Thus PRv = Pv follows, and the equality P:;' = pm Rv results immediately. (iv) In view of the embedding pm C pm Rv = P:;' the embedding R C Rv induces a homomorphism f : R/ pm ----+ Rv / P:;'. We shall now show that f is an isomorphism. Let a E R/ pm, a E a, and assume that a E Ker f.

Rai:::; Proof: (i) Let a be a non-zero algebraic integer and assume that 1. Let K = Q(a) and n = [K: Q]. The numbers a,a 2 , ••. all lie in RK, therefore their minimal polynomials have degrees not exceeding n. Since all conjugates of the numbers ak lie in the closed unit disc, the coefficients of their minimal polynomials do not exceed max{ (j) : j = 1, 2, ... , n }. • contains only finitely many distinct terms, and so for certain i -=f. e. a is a root of unity. (ii) Let a be a non-zero totally real integer whose conjugates all lie in the interval [-2, 2].