A refinement of Stark's conjecture over complex cubic number fields

We study the first-order zero case of Stark's conjecture over a complex cubic number field F. In that case, the conjecture predicts the absolute value of a complex unit in an abelian extension of F. We present a refinement of Stark's conjecture by proposing a formula (up to a root of unity) for the unit itself instead of its absolute value.     

On equivalence of number fields

The online version of the original article can be found at     

On equivalence of number fields

LetK be a field,G a finite group.G is calledK-admissible iff there exists a finite dimensionalK-central division algebraD which is a crossed product forG. Now letK andL be two finite extensions of the rationalsQ such that for every finite groupG, G isK-admissible if and only ifG isL-admissible. ThenK andL have the same degree and the same normal closure overQ. An erratum to this article is available at .     

On the first factor of the class number of prime cyclotomic fields

Let H(l) be the first factor of the class number of the field $\text{Q}$(exp 2πi/l), l a prime. The best-known upper and lower bounds on H(l) are improved for small l. The methods would also improve the best-known bounds for large l. It is shown that H(l) is the absolute value of the determinant of an easily written down matrix whose only entries are 0 and 1. The upper bounds obtained on H(l) significantly improve the Hadamard bound on the determinant of this matrix. Results of Lehmer on the factors of H(l) are explained via class field theory.     

On the number of quintic fields

We show that the number of quintic number fields whose discriminant does not exceed X in absolute value is bounded by a constant times X¹⁺ for any >0. This may be compared with the conjecture that the number of such fields is asymptotic to a constant times X as X tends to infinity.     

On the least primitive root in number fields

Let K be an algebraic number field and $ \mathfrak{O} $ _K its ring of integers. For any prime ideal $ \mathfrak{p} $ , the group $ (\mathfrak{O}_K /\mathfrak{p})* $ of the reduced residue classes of integers is cyclic. We call any element of a generator of the group $ (\mathfrak{O}_K /\mathfrak{p})* $ a primitive root modulo $ \mathfrak{p} $ . Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal $ \mathfrak{p} $ under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.     

On the large sieve in algebraic number fields

In the present note Bombieri's central theorem concerning the average distribution of the prime numbers in arithmetic progressions is generalized to arbitrary algebraic number fields.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 673–680, December, 1967.Finally, I express my profound gratitude to B. V. Levin for setting the problem and the help he rendered and to A. I. Vinogradov for valuable suggestions.     

On sign determinations in real algebraic number fields

Summary Given an ordered fieldF and a finite ordered extensionE. If it is possible to perform constructively the four rational operations inF and to determine the sign for an arbitrary element ofF, a rule is given for the sign determination of an arbitrary element ofE.     

On the parity of the class number of multiquadratic number fields

This paper investigates the 2-class group of real multiquadratic number fields. Let p₁,p₂,…,pn be distinct primes and . We draw a list of all fields K whose 2-class group is trivial.     

On solving relative norm equations in algebraic number fields

Let be algebraic number fields and a free -module. We prove a theorem which enables us to determine whether a given relative norm equation of the form has any solutions at all and, if so, to compute a complete set of nonassociate solutions. Finally we formulate an algorithm using this theorem, consider its algebraic complexity and give some examples.

     

On the distribution of ideals in cubic number fields

LetK be a cubic number field. Denote byA _K (x) the number of ideals with ideal norm ≤x, and byQ _K (x) the corresponding number of squarefree ideals. The following asymptotics are proved. For every ε>0 ε>0 $$\begin{gathered} {\text{ }}A_K (x) = c_1 x + O(x^{43/96 + \in } ), \hfill \\ Q_K (x) = c_2 x + O(x^{1/2} \exp {\text{ }}\{ - c(\log {\text{ }}x)^{3/5} (\log \log {\text{ }}x)^{ - 1/5} \} ). \hfill \\ \end{gathered}$$ Herec ₁,c ₂ andc are positive constants. Assuming the Riemann hypotheses for the Dedekind zeta function ζ _K , the error term in the second result can be improved toO(x ^53/116+ε).     

On the number of Diophantine m-tuples in finite fields

We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine m-tuples in finite fields, which is due to A. Dujella and M. Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).