Conjugate algebraic integers in real point sets

Mathematische Zeitschrift -     

Size of algebraic integers

Translated from Matematicheskie Zametki, Vol. 49, No. 4, pp. 152–154, April, 1991.     

Algebraic integers on the unit circle

By computing the rank of the group of unimodular units in a given number field, we provide a simple proof of the classification of the number fields containing algebraic integers of modulus 1 that are not roots of unity.     

On algebraic K-theory of real algebraic varieties with circle action

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S¹-action so that the quotient Y=X/S¹ is also a real algebraic variety. If π : X→Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with ,

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is onto, where , denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that

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for any real algebraic variety X. As an application we will show that for a compact connected Lie group G .     

Irreducible algebraic integers in short intervals

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x ^θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x ₀. The constant x ₀ effectively depends on K and θ. The author was supported in part by the grant no. N N201 1482 33 from the Polish Ministry of Science and Higher Education.     

Minimal solvable algebraic groups with a finite center

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 120–124, August, 1991.     

On the distribution of certain algebraic integers

Mathematische Zeitschrift -     

Totally real algebraic integers in small intervals

We show that there are irreducible monic polynomials having all roots in an interval of length close to 4. These are obtained by perturbing the coefficients of the respective Chebyshev polynomials. In particular, we obtain that our earlier lower bound for the house of totally real algebraic integers is sharp up to a logarithmic factor. Partially supported by the Lithuanian State Science and Studies Foundation. Published in Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 307–312, July–September, 2000.     

Hochschild homology of rings of algebraic integers

The purpose of this paper is to remind that the André-Quillen homology theory is a very useful tool to study Hochschild homology of commutative algebras. We choose as an example a computation of Hochschild homology of Dedekind extensions.     

On the distribution of irreducible algebraic integers

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2.     

On the distribution of irreducible algebraic integers

We study large values of the remainder term in the asymptotic formula for the number of irreducible integers in an algebraic number field K. In the case when the class number h of K is larger than 1, under certain technical condition on multiplicities of non-trivial zeros of Hecke L-functions, we detect oscillations larger than what one could expect on the basis of the classical Littlewood’s omega estimate for the remainder term in the prime number formula. In some cases the main result is unconditional. It is proved that this is always the case when h = 2. Author’s address: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614 Poznań, Poland The author was supported in part by KBN Grant # N N201 1482 33.     

On the partial quotients of algebraic integers

It is shown that there are algebraic integers, with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Davenport for algebraic numbers.     

Generating sets for rings of algebraic integers

Let R be the ring of integers of some finite algebraic extension of the rationals Q of degree n. A necessary and sufficient condition for s elements of R to be an R-basis is given, in terms of the Hermite normal form of a certain n ×ns integral matrix depending on the elements, and on the structure constants of R.     

The trace of totally positive algebraic integers

For all totally positive algebraic numbers except a finite number of explicit exceptions, the following inequality holds:

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where is the degree of and its conjugates. This improves previous results of Smyth, Flammang and Rhin.