Units in some families of algebraic number fields

Multi-dimensional continued fractions associated with are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature (2,1).

     

The minimum discriminant of sixth degree totally complex algebraic number fields

Inspired by previous results of Kaur, Hunter, and Mayer we developed a new method of determining the minimum value of the discriminant in absolute value of totally complex sixth degree algebraic number fields. It is 9747 and is attained only by the algebraic number field Q(θ), where $\text{θ =}\text{(2ω}{}^2\text{+ 5ω)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{? 1)}\text{(ω ? 1))}$ is a root of the monic irreducible polynomial x⁶ ? 3x⁵ + 4x⁴ ? 4x³ + 4x² ? 2x + 1 with discriminant ?9747 and ω is a primitive third root of unity.     

The polylogarithm in algebraic number fields

Based on Kummer's 2-variable functional equations for the second through fifth orders of the polylogarithm function, certain linear combinations, with rational coefficients, of polylogarithms of powers of an algebraic base were discovered to possess significant mathematical properties. These combinations are designated “ladders,” and it is here proved that the ladder structure is invariant with order when the order is decreased from its permissible maximum value for the corresponding ladder. In view of Wechsung's demonstration that the functions of sixth and higher orders possess no functional equations of Kummer's type, this analytical proof is currently limited to a maximum of the fifth order. The invariance property does not necessarily persist in reverse—increasing the order need not produce a valid ladder with rational coefficients. Nevertheless, quite a number of low-order ladders do lend themselves to such extension, with the needed additional rational coefficients being determined by numerical computation. With sufficient accuracy there is never any doubt as to the rational character of the numbers ensuing from this process. This method of extrapolation to higher orders has led to many quite new results; although at this time completely lacking any analytical proof. Even more astonishing, in view of Wechsung's theorem mentioned above, is the fact that in some cases the ladders can be validly extended beyond the fifth order. This has led to the first-ever results for polylogarithms of order six through nine. A meticulous attention to the finer points in the formulas was necessary to achieve these results; and a number of conjectural rules for extrapolating ladders in this way has emerged from this study. Although it is known that the polylogarithm does not possess any relations of a polynomial character with rational coefficients between the different orders, such relations do exist for some of the ladder structures. A number of examples are given, together with a representative sample of ladders of both the analytical and numerically-verified types. The significance of these new and striking results is not clear, but they strongly suggest that polylogarithmic functional equations, of a more far-reaching character than those currently known, await discovery; probably up to at least the ninth order.     

Character sums in algebraic number fields

The Pólya-Vinogradov inequality is generalized to arbitrary algebraic number fields K of finite degree over the rationals. The proof makes use of Siegel's summation formula and requires results about Hecke's zeta-functions with Grössencharacters. One application is to the problem of estimating a least totally positive primitive root modulo a prime ideal of K, least in the sense that its norm is minimal.     

Least quadratic non-residues in algebraic number fields

Let νp denote a totally positive integer of an algebraic number field K such that νp is a least quadratic non-residue modulo a prime ideal p of K, least in the sense that N(νp) is minimal. Then the following result is shown: For x ≥ 2 and ε > 0,

$\text{|p;Np?x}\text{and}\text{N(vp)>Np}{}^{\mathrm{\epsilon }}\text{|=O}{}_{\mathrm{\epsilon }}\text{(}\text{log log}\text{3x).}$

     

Permutation polynomials over algebraic number fields

Let K be an algebraic number field. It is known that any polynomial which induces a permutation on infinitely many residue class fields of K is a composition of cyclic and Chebyshev polynomials. This paper deals with the problem of deciding, for a given K, which compositions of cyclic or Chebyshev polynomials have this property. The problem is reduced to the case where K is an Abelian extension of $\text{Q}$. Then the question is settled for polynomials of prime degree, and the Abelian case for composite degree polynomials is considered. Finally, various special cases are dealt with.     

Cubic polynomials over algebraic number fields

We prove that every cubic form in 16 variables over an algebraic number field represents zero, generalizing the corresponding result of Davenport for cubic forms over the rationals. (This has already been proved for cubic forms in 17 or more variables by Ryavec.) We present this result as a special case of a “local-implies-global” theorem for cubic polynomials.     

Normal binomials over algebraic number fields

In this paper, normal and weakly normal binomials over an arbitrary algebraic number field will be characterized. Explicit results on the possible degrees of such binomials are given. Several examples conclude the paper.     

Approximation of number π by algebraic numbers from special fields

The estimate from below of the modulus of the difference between π and algebraic numbers from the fields generated by the roots of unity is made.     

Two-primary algebraic K-theory of two-regular number fields

We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order. Received August 1, 1998     

Conductor ideals of orders in algebraic number fields

We will give a new version of proof for Furtwängler’s characterization of all ideals of an algebraic number field which appear as conductors of orders of the field.