Complex Pisot numbers in algebraic number fields

Abstract

Let P (K ) be the set of Pisot numbers generating a real algebraic number field K over the field of rationals Q. Then, a result of Meyer implies that P (K ) is relatively dense in the interval [1, ∞) and a theorem of Pisot gives that P (K ) contains units, whenever K = Q.In the present note, we prove analogous results for the set of complex Pisot numbers generating a non-real number field K over Q when K is neither a quadratic field nor a CM-field.

On Number Fields Without a Unit Primitive Element

“…, d}. Then, as in the proof of Theorem 1.3(ii) (see [3]), the conjugates of α over R K are α and α, and the conjugates of the algebraic integer β := |α| 2 ∈ R K (over Q) are the numbers |α…”

“…Since K is cyclotomic when the degree of ζ is 2d, and since there are at most a finite number of cyclotomic fields with a given degree, we immediately see that the degree of ζ is not 2d when D is sufficiently small. Notice also that a calculation similar to the one in the proof of Corollary 1.7 shows that the degree of ζ cannot be equal to 2 when {i, i Using Corollary 3.1, one can easily deduce a positive answer to Question 1.8 when R runs through the normal cubic fields C n := Q(θ n ) defined by the conditions θ 3 n − nθ n + n = 0, where n is a square-free positive rational integer, the residue (mod 3) of each prime divisor of n is 1 and 4n − 27 is the square of a rational integer.…”

“…It is clear that we have a negative answer to Question 1.2 when K = Q or when K is a nonreal quadratic field, since the units of K, in these two cases, belong to Ω K . The following theorems collect some known answers to Question 1.2; the first one is due to Pisot (see for instance [2,5,13]) and the other may easily be deduced from the results of [3]. Theorem 1.3.…”

On Real Parts of Powers of Complex Pisot Numbers

“…A complex Salem number is a nonreal algebraic integer with modulus greater than 1 whose other conjugates, except its complex conjugate, are of modulus at most 1, and having a conjugate with modulus 1. Recently, Bertin and the author [3] have shown that the family of complex Pisot numbers generating a nonreal number field is a complex Meyer set. Another of Pisot's results about the distribution modulo one of powers of real numbers asserts that if the set, say L = L(θ, λ), of limit points of the sequence ({λθ n }) n∈N , where λ is a nonzero real number and θ is a real algebraic number greater than 1, has a finite number of limit points, then θ ∈ S and λ ∈ Q(θ) [16].…”

“…Let θ be a complex Pisot number or a complex Salem number. Then, by the result of [3] mentioned above, there is a complex Pisot number, say α, generating the field Q(θ). Let ε > 0 and let N 1 ∈ N be such that 3≤ j≤d |α j | N 1 < ε.…”