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Recall that for a domain R where every nonzero nonunit factors into irreducibles, the elasticity of R is defined as sup ⁡ { s r : π 1 ⋯ π r = ρ 1 ⋯ ρ s , with all π i , ρ j irreducible }. We call a quadratic field K maximally elastic if the ring of integers of K is a UFD and each element of { 1 , 3 2 , 2 , 5 2 , 3 , ... } ∪ { ∞ } appears as an elasticity of infinitely many orders inside K. This corresponds to the orders in K exhibiting, to the extent possible for a quadratic field, maximal variation in terms of the failure of unique factorization. Assuming the Generalized Riemann Hypothesis, we prove that K = Q (2) is universally elastic, and we provide evidence for a conjectured characterization of maximally elastic quadratic fields. [ABSTRACT FROM AUTHOR]

Recall that D is a half-factorial domain (HFD) when D is atomic and every equation π 1 ⋯ π k = ρ 1 ⋯ ρ ℓ , with all π i and ρ j irreducible in D, implies k = ℓ . We explain how techniques introduced to attack Artin's primitive root conjecture can be applied to understand half-factoriality of orders in real quadratic number fields. In particular, we prove that (a) there are infinitely many real quadratic orders that are half-factorial domains, and (b) under the generalized Riemann hypothesis, Q (2) contains infinitely many HFD orders. [ABSTRACT FROM AUTHOR]

Here, we study the action of the groups H (λ) generated by the linear fractional transformations x : z ↦ − 1 z  and  w : z ↦ z + λ and λ is a positive integer, on the subsets ℚ ∗ (n) = { a + n c | a , b = a 2 − n c , c ∈ ℤ } ∪ { 0 , 1 , ∞ } and | n | is a square-free positive integer. We prove that this action has a finite number of orbits if and only if λ = 1 or λ = 2. Moreover, we give an upper bound for the number of orbits for λ = 2. [ABSTRACT FROM AUTHOR]

We study the structure of nets over quadratic fields. Let be a quadratic field, and let be the ring of integers of . A set of additive subgroups of is a net (carpet) of order over if for all values of the indices , , , . A net is irreducible if all additive subgroups are nonzero. A net is a -net if , . Let be an irreducible -net of order over , where are -modules. We prove that, up to conjugation by a diagonal matrix, all are fractional ideals of a fixed intermediate subring , , and all diagonal rings coincide with ; i.e., , where are integer ideals of for all , and if . Furthermore, for all and . [ABSTRACT FROM AUTHOR]

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Let ℚ be the rational filed. For a square-free integer d with d ≠ 0 , 1 , we denote by K = ℚ (d) the quadratic field. Let K be the ring of algebraic integers of K. In this paper, we completely determine the unit group of the quotient ring K / 〈 ξ n 〉 of K for an arbitrary prime ξ in ℚ (d) , where ℚ (d) has the unique factorization property, and n > 1 is a rational integer. [ABSTRACT FROM AUTHOR]

Let K be a quadratic field over the rational field and a K (n) be the number of nonzero integral ideals with norm n. We establish Erdős-Kac type theorems weighted by a K (n) l and a K (n 2) l of quadratic field in short intervals with l ∈ ℤ+. We also get asymptotic formulae for the average behavior of a K (n) l and a K (n 2) l in short intervals. [ABSTRACT FROM AUTHOR]

In this paper we consider the set of Mahler measures of generalized Pisot and Salem numbers and give a characterization of such numbers in terms of their minimal polynomials. We show that every real quadratic algebraic integer β can be expressed by the difference of the Mahler measure of a generalized Pisot number and a Pisot number in ℚ(β). [ABSTRACT FROM AUTHOR]

In this paper, the authors show that there exists infinitely many family of pairs of quadratic fields ℚ (D) and ℚ (D + n) with D,n ∈ ℤ whose class numbers are both divisible by 3. [ABSTRACT FROM AUTHOR]

We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for x ≥ 2 . Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at s = 1 . [ABSTRACT FROM AUTHOR]

Using elementary methods, we count the quadratic residues of a prime number of the form p = 4 n − 1 in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number h of the imaginary quadratic field Q (− p). Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found. [ABSTRACT FROM AUTHOR]

In this paper, we consider the solvability of the Pellian equation x 2 − (d 2 + 1) y 2 = − m , $$\begin{array}{} \displaystyle x^2-(d^2+1)y^2 = -m, \end{array} $$ in cases d = nk, m = n2l−1, where k, l are positive integers, n is a composite positive integer and d = pq, m = pq2, p, q are primes. We use the obtained results to prove results on the extendibility of some D(−1)-pairs to quadruples in the ring Z [ − t ] $\begin{array}{} {\mathbb{Z}}[\sqrt{-t}] \end{array} $ , with t > 0. [ABSTRACT FROM AUTHOR]

For L / K a finite extension of algebraic number fields, L may or may not have a relative integral basis over K. We show the existence of relative integral basis of a biquadratic field L = Q (m , n) over its quadratic subfield K = Q (m) is equivalent to that K is a Pólya field, or equivalently all strongly ambiguous ideal classes of K are trivial. [ABSTRACT FROM AUTHOR]

We investigate the Diophantine equation of the form , with where and are odd and is even, and is a sufficiently small natural number. We obtain a complete description of the set of solutions to such an equation. [ABSTRACT FROM AUTHOR]

We define a gap function to measure the difference of two distinct prime ideals in a given number field. In this paper, we determine all quadratic fields and cyclotomic fields satisfying the condition: There exist two distinct prime ideals whose gap is 1.

To prevent an exhaustive key-search attack of the key-exchange protocol using real quadratic fields, we need to ensure that the number ` of reduced principal ideals in K is sufficiently large. In this paper we present an example of a family which are not valid for this protocol. [ABSTRACT FROM AUTHOR]

Let n > 1 be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form ℚ ( x 2 − 2 y n ) whose ideal class group has an element of order n. This family gives a counterexample to a conjecture by H. Wada (1970) on the structure of ideal class groups. [ABSTRACT FROM AUTHOR]

For a given odd positive number n and a positive integer m, we show that there exist infinitely many pairs of imaginary quadratic fields Q (D) and Q (D + m) with D ∈ Z whose class groups have an element of order n. [ABSTRACT FROM AUTHOR]

We study the extendibility of a D (- 1) -pair {1, p}, where p is a Fermat prime, to a D (- 1) -quadruple in Z [ - t ] , t > 0 . [ABSTRACT FROM AUTHOR]

In this paper, we prove that the equation x 2 - (p 2 k + 2 + 1) y 2 = - p 2 l + 1 , l ∈ { 0 , 1 , ⋯ , k } , k ≥ 0 , where p is an odd prime number, is not solvable in positive integers x and y. By combining that result with other known results on the existence of Diophantine quadruples, we are able to prove results on the extensibility of some D (- 1) -pairs to quadruples in the ring Z [ - t ] , t > 0 . [ABSTRACT FROM AUTHOR]

In this study, we construct the efficiently computable endomorphisms on elliptic curves with j-invariant 1728, to accelerate the computation of ISD method. The ISD method computed scalar multiplication on elliptic curves where it requires three endomorphisms to accomplish. However, the original ISD method only able to solve integer multiplications since their endomorphisms are defined over Z. Besides, the endomorphisms defined in the original ISD method are not efficiently computable. We extend the study by defining the endomorphisms in the ISD method over the Q(√-d), so that it can solve complex multiplications. Elliptic curves with j-invariant 1728 are defined over Q(i), where its discriminant is given as D = -4, with a unique maximal order. The maximal order satisfies a polynomial of degree two, which represents the minimal polynomial for the first efficiently computable endomorphism. Meanwhile, we choose the other two endomorphisms to belong to Q(i) as well. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the cyclicity of the 2 -class group of the first Hilbert 2 -class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let p 1 ≡ p 2 ≡ − q ≡ 1 (mod 4) be different prime integers. Put 𝕜 = ℚ ( p 1 p 2 q) , and denote by C 𝕜 , 2 its 2 -class group and by 𝕜 2 (1) (respectively 𝕜 2 (2) ) its first (respectively second) Hilbert 2 -class field. Then, we are interested in studying the metacyclicity of G = Gal (𝕜 2 (2) / 𝕜) and the cyclicity of Gal (𝕜 2 (2) / 𝕜 2 (1)) whenever the 4 -rank of C 𝕜 , 2 is 1. [ABSTRACT FROM AUTHOR]

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\Bbbk=\Bbb Q(\sqrt{2pq}, {\rm i})$, where ${\rm i}=\sqrt{-1}$ and $p\equiv-q\equiv1 \pmod4$ are different primes. For each of the three quadratic extensions $\Bbb K/\Bbbk$ inside the absolute genus field $\Bbbk^{(*)}$ of $\Bbbk$, we determine a fundamental system of units and then compute the capitulation kernel of $\Bbb K/\Bbbk$. The generators of the groups ${\rm Am}_s(\Bbbk/F)$ and ${\rm Am}(\Bbbk/F)$ are also determined from which we deduce that $\Bbbk^{(*)}$ is smaller than the relative genus field $(\Bbbk/\Bbb Q({\rm i}))^*$. Then we prove that each strongly ambiguous class of $\Bbbk/\Bbb Q({\rm i})$ capitulates already in $\Bbbk^{(*)}$, which gives an example generalizing a theorem of Furuya (1977).

This note is devoted to a solution of a problem posed by Ladislav Skula. [ABSTRACT FROM AUTHOR]