Norm form equations and continued fractions.
Norm limitation theorems of class field theory.
Norm residue symbol and cyclotomic units
Normal bases for the space of continuous functions defined on a subset of Zp.
Let K be a non-archimedean valued field which contains Qp and suppose that K is complete for the valuation |·|, which extends the p-adic valuation. Vq is the closure of the set {aqn|n = 0,1,2,...} where a and q are two units of Zp, q not a root of unity. C(Vq → K) is the Banach space of continuous functions from Vq to K, equipped with the supremum norm. Our aim is to find normal bases (rn(x)) for C(Vq → K), where rn(x) does not have to be a polynomial.
Normal bases of rings of continuous functions constructed with the (qₙ)-digit principle
Normal basis and Greenberg' s conjecture.
Normal bivariate Birkhoff interpolation schemes and Pell equation
Finding the normal Birkhoff interpolation schemes where the interpolation space and the set of derivatives both have a given regular “shape” often amounts to number-theoretic equations. In this paper we discuss the relevance of the Pell equation to the normality of bivariate schemes for different types of “shapes”. In particular, when looking at triangular shapes, we will see that the conjecture in Lorentz R.A., Multivariate Birkhoff Interpolation, Lecture Notes in Mathematics, 1516, Springer, Berlin-Heidelberg,...
Normal integral bases and complex conjugation.
Normal Integral Bases and Embedding Problems.
Normal integral bases and ray class groups
Normal integral bases and tameness conditions for Kummer extensions
We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer extensions...
Normal integral bases and the Galois groups of Galois extensions of number fields.
Normal integral bases and the Spiegelungssatz of Scholz
Normal integral bases for Emma Lehmer’s parametric family of cyclic quintics
Explicit normal integral bases are given for some cyclic quintic fields defined by Emma Lehmer’s parametric family of quintics.
Normal integral bases for infinite abelian extensions
Normal number constructions for Cantor series with slowly growing bases
Let be a sequence of bases with . In the case when the are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose -Cantor series expansion is both -normal and -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of , and from this construction we can provide computable constructions of numbers with atypical normality properties.
Normal numbers and the Borel hierarchy
We show that the set of absolutely normal numbers is Π⁰₃-complete in the Borel hierarchy of subsets of real numbers. Similarly, the set of absolutely normal numbers is Π⁰₃-complete in the effective Borel hierarchy.
Normal numbers and the middle prime factor of an integer
Let pₘ(n) stand for the middle prime factor of the integer n ≥ 2. We first establish that the size of log pₘ(n) is close to √(log n) for almost all n. We then show how one can use the successive values of pₘ(n) to generate a normal number in any given base D ≥ 2. Finally, we study the behavior of exponential sums involving the middle prime factor function.
Normal order for a function associated with factorization into irreducibles
Normal orders for certain functions associated with factorizations in number fields