1 Algebraic number fields 3 1.1 Algebraic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Algebraic number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Norm and Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Algebraic integers . . . . . . . . . . . . . . . . . . . . . . . . ....

We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston [Thu].

We shall extend the results of [5] and prove that if f = Z o a x ? Z [[X]] is algebraic over Q (x), where a = 1, ƒ 1 and if ? , ? ,..., ? are p-adic integers, then 1 ? , ? ,..., ? are linkarly independent over Q if and only if (1+x) ,(1+x) ,…,(1+x) are algebraically independent over Q (x) if and only if f , f ,.., f are algebraically independent over Q (x)

we shall extend the results of [5] and prove that if f = z o a x ? z [[x]] is algebraic over q (x), where a = 1, ƒ 1 and if ? , ? ,..., ? are p-adic integers, then 1 ? , ? ,..., ? are linkarly independent over q if and only if (1+x) ,(1+x) ,…,(1+x) are algebraically independent over q (x) if and only if f , f ,.., f are algebraically independent over q (x)

Let α be a reciprocal algebraic integer of degree d. The house is the largest modulus its conjugates. We compute minimum houses all integers d having minimal polynomial which factor Dth or antireciprocal with at most eight monomials, say mr(d), for 180, D≤1.5d and D≤210. show that it not necessary to take into account imprimitive polynomials. computations suggest several conjectures. d-th power...