This paper provides the first examples of rings of algebraic numbers containing the rings of algebraic integers of the infinite algebraic extensions of Q, where Hilbert’s Tenth Problem is undecidable.

Algebraic Number Theory: • What is it? The goals of the subject include: (i) to use algebraic concepts to deduce information about integers and other rational numbers and (ii) to investigate generalizations of the integers and rational numbers and develop theorems of a more general nature. Although (ii) is certainly of interest, our main point of view for this course will be (i). The focus of t...

Recently, discrete sets of numbers, the-integers ZZ , have been proposed as numbering tools in quasicrystalline studies. Indeed, there exists a unique numeration system based on the irrational > 1 in which the-integers are all real numbers with no fractional part. These-integers appear as being quite appropriate to describing some quasilattices relevant to quasicrystallography when precisely is...

We classify a quasiperiodic flow as either algebraic or transcendental. For an algebraic quasiperiodic flow φ on the n-torus, Tn, we prove that an absolute invariant of the smooth conjugacy class of φ, known as the multiplier group, is a subgroup of the group of units of the ring of integers in a real algebraic number field F of degree n over Q. We also prove that for any real algebraic number ...

In analogy with the Riemann zeta function at positive integers, for each finite field Fpr with fixed characteristic p we consider Carlitz zeta values ζr(n) at positive integers n. Our theorem asserts that among the zeta values in ∪∞ r=1 {ζr(1), ζr(2), ζr(3), . . . }, all the algebraic relations are those relations within each individual family {ζr(1), ζr(2), ζr(3), . . . }. These are the algebr...

As an analogue to special values at positive integers of the Riemann zeta function, for each constant field Fpr with fixed characteristic p we consider Carlitz zeta values ζr(n) at positive integers n. The main theorem of this paper asserts that among the families of Carlitz zeta values ∪∞ r=1 {ζr(1), ζr(2), ζr(3), . . . }, all the algebraic relations are those algebraic relations among each in...