On a Special Class of Reduced Algebraic Numbers

Citation

Wolfskill, John Charles (1982) On a Special Class of Reduced Algebraic Numbers. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5jj1-j729. https://resolver.caltech.edu/CaltechETD:etd-09082006-132532

Abstract

The notion of a reduced real quadratic number goes back to Gauss, who defined such a number to be reduced if it is greater than one, and its conjugate between negative one and zero. An equivalent characterization is that the continued fraction of a reduced quadratic number is purely periodic. Zassenhaus generalized this by defining a real algebraic number α to be reduced if α > 1 and -1 < Reα' < 0 for the conjugates α' of α distinct from α. In this thesis, several properties of these reduced numbers are developed. In particular it is shown that there exist reduced numbers α with the property that α has no reduced immediate predecessor, that is, u + 1/α is not reduced for any choice of the rational integer u. We call such a number α an ancestor. These ancestors have the property that every real algebraic number of degree at least three is equivalent to exactly one of them. Here, equivalence is in the sense of continued fractions; α ~ β means that there exist integers a, b, c, and d such that ad - bc = ±1 and α = aβ+b/cβ+d. This is equivalent to α and β having identical continued fractions after a certain point. This property of ancestors gives rise to an application to the problem of determining whether or not two given integral binary homogeneous forms are equivalent, assuming that each form has a real root. If the forms are equivalent, so are the roots of the forms; this can be checked by comparing the ancestors. This method is computationally effective.

In another direction, there is a connection between the reduced numbers defined above and the Pisot-Vijayaraghavan (PV) numbers (a PV number is a real algebraic integer greater than one all of whose other conjugates have absolute value less than one). It turns out that any reduced algebraic integer which is not an ancestor is a PV number; integral ancestors may or may not be. Part of the thesis is devoted to a more detailed comparison of PV numbers and integral ancestors. On one side, there is the theorem of Salem that the PV numbers are closed. On the other, it is proved here that if K is a field of degree at least three over the rationals, real but not totally real, then no integral ancestor in K is isolated (that is, there are other integral ancestors arbitrarily close). Much more is true; one can show in many cases that the integral ancestors in such a field lie in a set of non-trivial intervals in which they are dense. This decomposition is studied in more detail. For example, in Q(α), where α³ = α + 1, the integral ancestors are actually dense in [1,∞]. In contrast, in Q(∛2), the integral ancestors are dense in [1,2] U [3,5] U [6,8] U [9,11] U . . . and none of them occur in the gaps. It is proved that all cubic fields which are not totally real are like one of these two fields in the way the integral ancestors are distributed. Similar results hold for fields of higher degree, although the situation is somewhat more complicated.

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