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We present two families of numerical semigroups and show that for each family, the number of required components in an irreducible decomposition cannot be bounded by any given integer. This gives a negative answer to a question raised by…

Let $K$ be a number field with ring of integers $\mathbb{Z}_K$. We prove two asymptotic formulas connected with the distribution of irreducible elements in $\mathbb{Z}_K$. First, we estimate the maximum number of nonassociated irreducibles…

We give an explicit version of a result due to D. Burgess. Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $\chi$ takes on a particular value is less…

Let $n (>3)$ be a prime number and $\Bbb F_{2^n}$ a finite field of $2^n$ elements. Let $L =\Bbb F_{2^n}\cup \{\infty\}$ be the support set and $g(x)$ an irreducible polynomial of degree $6$ over $\Bbb F_{2^n}$. In this paper, we obtain an…

We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative…

We provide explicit bounds for the number of integral ideals of norms at most $X$ is $\mathbb{Q}[\sqrt{d}]$ when $d <0$ is a fundamendal discriminant with an error term of size $O(X^{1/3})$. In particular, we prove that, when $\chi$ is the…

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as…

The roots of -1 in the set of biquaternions (quaternions with complex components, or complex numbers with quaternion real and imaginary parts) are studied and it is shown that there is an infinite number of non-trivial complexified…

Let $\mathcal{O}$ be an order in an algebraic number field and suppose that the set of distances $\Delta(\mathcal{O})$ of $\mathcal{O}$ is nonempty (equivalently, $\mathcal{O}$ is not half-factorial). If $\mathcal{O}$ is seminormal (in…

In this paper we study practical numbers of some special forms. For any integers $b\ge0$ and $c>0$, we show that if $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many nonnegative integers $n$ with $n^2+bn+c$…

We study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras. We consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension. Let $B$…

There are four characteristic circles for each triangle on a plane. All for are tangential to the three straight lines containing the triangles' three sides. Three are exterior circles, the fourth is the in-circle. When the triangle is…

Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$ with $\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive from a result of Hoa…

The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as…

Let $d\geq 2$ be a squarefree integer, let $\omega\in\{\sqrt{d},\frac{1+\sqrt{d}}{2}\}$ be such that $\mathbb{Z}[\omega]$ is the ring of algebraic integers of the real quadratic number field $\mathbb{Q}(\sqrt{d})$, let $\varepsilon>1$ be…

In this paper, we determine the 2-rank of the class group of certain classes of real cyclic quartic number fields. Precisely, we consider the case in which the quadratic subfield is Q(\sqrt{l}) with l=2 or a prime congruent to 1 mod 8.

We study the capitulation of ideal classes in an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $k =Q(\sqrt{2pq}, i)$, where $i=\sqrt{-1}$ and $p\equiv -q\equiv1 \pmod 4$ are different primes. For each…

This paper continues arXiv.org:math.AG/0609256 and arXiv:0708.3991 Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at…

For certain types of quadratic forms lying in the n-th power of the fundamental ideal, we compute upper bounds and where possible exact values for the minimal number of general n-fold Pfister forms, that are needed to write the Witt class…

Let $R$ be a finite ring and define the hyperbola $H=\{(x,y) \in R \times R: xy=1 \}$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following "square root law" bound holds with a constant $C>0$ for…