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Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q . Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q . In this paper, we generalise the torsor method for integral points from Q to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A 3 over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin's conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework. [ABSTRACT FROM AUTHOR]

It is well known that factorization is not unique in Z [ − 5 ] . We give a short, self-contained proof that Z [ − 5 ] is "halfway" toward being a unique factorization domain: For every nonzero, nonunit α ∈ Z [ − 5 ] , any two factorizations of α into irreducibles involve the same number of factors. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for a given positive integer k, there are at least x 1 / 3 - o (1) integers d ≤ x such that the consecutive pure cubic fields Q (d + 1 3) , ⋯ , Q (d + k 3) have arbitrarily large class numbers. [ABSTRACT FROM AUTHOR]

A family of fractal arrangements of circles is introduced for each imaginary quadratic field K. Collectively, these arrangements contain (up to an affine transformation) every set of circles in the extended complex plane with integral curvatures and Zariski dense symmetry group. When that set is a circle packing, we show how the ambient structure of our arrangement gives a geometric criterion for satisfying the asymptotic local–global principle. Connections to the class group of K are also explored. Among them is a geometric property that guarantees certain ideal classes are group generators. [ABSTRACT FROM AUTHOR]

For any integer n > 1, we provide a parametric family of biquadratic fields with class groups having n-rank at least 2. Moreover, in some cases, the n-rank is bigger than 4. [ABSTRACT FROM AUTHOR]

Let p ≥ 5 be a prime number such that p ≡ 3 (mod 8) and d an odd positive squarefree integer. Put K = Q (2 , - p , d) . In this paper, we compute the 2-rank of the class group of the fields K = Q (2 , - p , d) . Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic and of type (2, 2) or (2, 4). [ABSTRACT FROM AUTHOR]

Let f = X 6 − 3 X 2 − 1 ∈ Q [ X ] and let L f be the splitting field of f over Q . We show by hand that the Galois group Gal (L f / Q) of the Galois extension L f / Q is isomorphic to the alternating group A4. Moreover, we show that the six roots of f correspond to the six edges of a tetrahedron and that the four roots of the polynomial X 4 + 18 X 2 − 72 X + 81 correspond to the four faces of a tetrahedron, which allows us to determine all eight proper intermediate fields of the extension L f / Q . [ABSTRACT FROM AUTHOR]

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number β . We prove that if β is real quadratic, then the number of partitions is always finite if and only if some conjugate of β is larger than 1. Further, we show that for β satisfying a certain condition, the partition function attains all non-negative integers as values. [ABSTRACT FROM AUTHOR]

Let S = { a 1 , a 2 , ⋯ , a n } be a finite set of non-zero integers. In [5], Karthick Babu and Anirban Mukhopadhyay calculated the explicit structure of the Galois group of multi-quadratic field Q (a 1 , a 2 , ⋯ , a n) over Q . For a positive integer d ⩾ 3 , ζ d denotes the primitive d-th root of unity. In this paper, we calculate the explicit structure of the Galois group of Q (a 1 , a 2 , ⋯ , a n , ζ d) over Q in terms of its action on ζ d and a i for 1 ⩽ i ⩽ n . [ABSTRACT FROM AUTHOR]

We give an elementary approach to studying whether rings of S-integers in complex quadratic fields are Euclidean with respect to the S-norm. [ABSTRACT FROM AUTHOR]

For an imaginary biquadratic number field K = Q (- q , d) , where q > 3 is a prime congruent to 3 (mod 8) , and d is an odd square-free integer which is not equal to q, let K ∞ be the cyclotomic Z 2 -extension of K . For any integer n ≥ 0 , we denote by K n the nth layer of K ∞ / K . We investigate the rank of the 2-class group of K n , then we draw the list of all number fields K such that the Galois group of the maximal unramified pro-2-extension over their cyclotomic Z 2 -extension is metacyclic pro-2 group. [ABSTRACT FROM AUTHOR]

We extend formulae of Mertens and Mirsky on the asymptotic behaviour of the usual Euler function to the Euler functions of principal rings of integers of imaginary quadratic number fields, giving versions in angular sectors and with congruences. [ABSTRACT FROM AUTHOR]

Codebooks with small maximum cross-correlation amplitudes are used to distinguish the signals from different users in multiple access communication systems in code division. Firstly, this paper studies the Jacobi sums over Galois rings of arbitrary characteristics and completely determines their absolute values. This extends the work by Li et al. (Sci China 56(7):1457–1465, 2013), where the Jacobi sums over some Galois rings with characteristics of a prime square were discussed. It is worth mentioning that the generalization is not trivial, as the Galois rings of arbitrary characteristics have a more complicated structure. Our deterministic construction of codebooks is based on Jacobi sums over Galois rings of arbitrary characteristics, producing asymptotically optimal codebooks for the Welch bound. Finally, compared to the literature, this article proposes for the first time design of codebooks based on Jacobi sums over Galois rings. In addition, the parameters of the presented codebooks are new. [ABSTRACT FROM AUTHOR]

The reduction algorithm is used to compute the reduced ideals of a number field. However, there are reduced ideals that can never be obtained from this algorithm. In this paper, we will show that these ideals have inverses of larger norms among reduced ones. Especially, for some number fields, we present the conditions of the reduced ideals produced by the reduction algorithm. Additionally, our results partly answer the question by R. Schoof who asks about the number of the reduced ideals that can not be produced by the reduction algorithm. [ABSTRACT FROM AUTHOR]

Given commutative, unital rings A and B with a ring homomorphism A → B making B free of finite rank as an A -module, we can ask for a "trace" or "norm" homomorphism taking algebraic data over B to algebraic data over A . In this paper we we construct a norm functor for the data of a quadratic algebra: given a locally-free rank-2 B -algebra D , we produce a locally-free rank-2 A -algebra Nm B / A (D) in a way that is compatible with other norm functors and which extends a known construction for étale quadratic algebras. We also conjecture a relationship between discriminant algebras and this new norm functor. [ABSTRACT FROM AUTHOR]

On the Waring's problem for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as sums of kth powers with $ k\in \{2,3,4,5,6,7,8\} $ k ∈ { 2 , 3 , 4 , 5 , 6 , 7 , 8 } in the literature. In this paper, we provide similar conditions for matrices written as sums of kth powers with $ k\in \{9,10,11,12,13,14,15,16\} $ k ∈ { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 }. [ABSTRACT FROM AUTHOR]

Previous authors have classified the possible factor rings of the Gaussian integers and the Eisenstein integers. Here, we extend this classification to the ring of integers of any quadratic number field, provided the ring has unique factorization. In the case of imaginary quadratic fields, the classification has exactly the same flavor as that of the Gaussian integers and Eisenstein integers. For real quadratic fields, the classification is only slightly more complicated. [ABSTRACT FROM AUTHOR]

In this article, using the arithmetic of Hurwitz zeta functions and Gauss sums, we investigate the arithmetic nature of ζ K (2 k + 1) ζ K + (2 k + 1) , where K is any totally imaginary abelian extension with maximal totally real subfield K + and k ≥ 1 is an integer. As a consequence, we derive an elementary proof of the elegant result of Murty and Pathak about irrationality of Dedekind zeta values associated to imaginary quadratic fields at odd integers. This alternate line of action allows us to link these questions to a question of Chowla and Milnor, resulting in some new results about Dedekind zeta values. [ABSTRACT FROM AUTHOR]

In this paper we discuss an analogue of Hilbert's theorems 105 and 106, i.e. a reinterpretation of Gauss' "genus theory", for imaginary quadratic fields of class number one. We explicitly compute all biquadratic fields with Hasse unit index one whose ambiguous ideal classes over an imaginary quadratic field of class number one are principal. [ABSTRACT FROM AUTHOR]

This paper deals with quadratic irrationals of the form m / q + v for fixed positive integers v and q, v not a square, and varying integers m, (m , q) = 1 . Two numbers m / q + v , n / q + v of this kind are equivalent (in a classical sense) if their continued fraction expansions can be written with the same period. We give a necessary and sufficient condition for the equivalence in terms of solutions of Pell's equation. Moreover, we determine the number of equivalence classes to which these quadratic irrationals belong. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there are infinitely many real quadratic fields with odd class number divisible by 3. [ABSTRACT FROM AUTHOR]

For any square-free positive integer m ≡ 10 (mod 16) with m ⩾ 26, we prove that the class number of the real cyclotomic field ℚ(ζ4m +ζ4m−1) is greater than 1, where ζ4m is a primitive 4mth root of unity. [ABSTRACT FROM AUTHOR]

In this article we study the lower bounds for the class numbers of certain families of totally real biquadratic fields. The bounds we have obtained represent an improvement over the previously known bounds provided by Mollin (Proc Am Math Soc 101(3):439–444, 1987). Additionally, we demonstrate that the obtained bounds can be further improved by a factor of two under certain mild assumptions. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d) for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2). [ABSTRACT FROM AUTHOR]

In this paper, we present an alternative method to compute all imaginary octic non-CM fields with class number one, listed in Yamamura’s paper (Acta Arith 86:133–47, 1998). We consider such octic fields as an abelian extension over an appropriate quadratic subfield, whereas Yamaura’s method uses a known structure for quartic fields. In fact, we demonstrate that all such imaginary non-CM octic fields with class number one are subfields of the ray class fields of some imaginary quadratic fields with an appropriate conductor. Using the Pari software, as described in Sect. 3, we determine all imaginary octic non-CM fields with class number one. (The codes are included as an appendix.) [ABSTRACT FROM AUTHOR]

As a consequence of their work, Bruce C. Berndt, Ronald J. Evans, Larry Joel Goldstein and Michael Razar obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind sums. We give a short proof of it and also express the relative class numbers of imaginary abelian number fields in terms of Dedekind sums. [ABSTRACT FROM AUTHOR]

A D(n)-m-tuple, where n is a non-zero integer, is a set of m distinct elements in a commutative ring R such that the product of any two distinct elements plus n is a perfect square in R. In this paper, we prove that there does not exist a D(−1)-quadruple { a , b , c , d } in the ring Z [ - k ] , k ≥ 2 with positive integers a < b < 16 a 2 - a - 2 + 2 k (8 a 2 + 3 a + 1) and integers c and d satisfying d < 0 < c . By combining that result with [14, Theorem 1.1] we were able to obtain a general result on the non-existence of a D(−1)-quadruple { a , b , c , d } in Z [ - k ] with integers a, b, c, d satisfying a < b ≤ 8 a - 3 . Furthermore, for a non-negative integer i and a positive integer j, we apply the obtained results in proving of the non-existence of D(−1)-quadruples containing powers of primes p i , q j with an arbitrary different primes p and q. [ABSTRACT FROM AUTHOR]

Let n be an integer ≥ 1 and p 1 ,... , p n distinct odd prime integers. In this article, we give the list of all imaginary multiquadratic number fields K n = Q (i , 2 , p 1 , ... , p n) that have a cyclic 2-class group. [ABSTRACT FROM AUTHOR]

For a metabelian 2-group G such that G / G ′ ≃ (2 , 2 n) , n ≥ 2 , we are interested in giving necessary and sufficient conditions to have G abelian, modular, metacyclic non modular or nonmetacyclic. The abelianizations of the subgroups of index 2 and 4, and the kernel of the transfer map in the metacyclic case are given too. As an application of these results, we investigate the capitulation of 2-classes of fields whose 2-class groups are of type (2 , 2 n) , n ≥ 2 . Examples are given by considering some real quadratic fields. [ABSTRACT FROM AUTHOR]

We study the class number one and class number two problems for real quadratic fields Q (9 m 2 + 4 m) , where m is an odd integer. We show that for m ≡ 1 (mod 3) there is only one such field with class number one and only one such field with class number two. Moreover, we also evaluate some generalized Dedekind sums. [ABSTRACT FROM AUTHOR]

In this paper, we show that every quadratic algebraic number lying in a quartic extension L of Q can be written as a quadratic polynomial with rational coefficients in a generator of L over Q . On the other hand, we prove that for each totally real algebraic number β of degree d ≥ 3 , there are infinitely many quadratic extensions L of Q (β) such that β cannot be expressed by a quadratic polynomial with rational coefficients in a generator of L over Q . The same assertion is proved for a large class of cubic algebraic numbers β without assumption that β is totally real. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. If an integer g generates a subgroup of index t in (Z / p Z) ∗ , then we say that g is a t-near primitive root modulo p. In this paper, for a subset { a 1 , a 2 , ⋯ , a n } of Z \ { - 1 , 0 , 1 } , we prove each coprime residue class contains a positive density of primes p not having a i as a t-near primitive root and with the a i satisfying a prescribed residue pattern modulo p, for 1 ≤ i ≤ n . We also prove a more refined variant of it. [ABSTRACT FROM AUTHOR]

An integral quadratic lattice is called indefinite k-universal if it represents all integral quadratic lattices of rank k for a given positive integer k. For k ≥ 3 , we prove that the indefinite k-universal property satisfies the local–global principle over number fields. For k = 2 , we show that a number field F admits an integral quadratic lattice which is locally 2-universal but not indefinite 2-universal if and only if the class number of F is even. Moreover, there are only finitely many classes of such lattices over F. For k = 1 , we prove that F admits a classic integral lattice which is locally classic 1-universal but not classic indefinite 1-universal if and only if F has a quadratic unramified extension where all dyadic primes of F split completely. In this case, there are infinitely many classes of such lattices over F. All quadratic fields with this property are determined. [ABSTRACT FROM AUTHOR]

Let d be a square-free positive integer and h(d) be the class number of the real quadratic field ℚ (d) . We give an explicit lower bound for h(n2 + r), where r = 1, 4. Ankeny and Chowla proved that if g > 1 is a natural number and d = n2g +1 is a square-free integer, then g ∣ h(d) whenever n > 4. Applying our lower bounds, we show that there does not exist any natural number n > 1 such that h(n2g + 1) = g. We also obtain a similar result for the family ℚ ( n 2 g + 4 ) . As another application, we deduce some criteria for a class group of prime power order to be cyclic. [ABSTRACT FROM AUTHOR]

Let d ≡ 2 (mod 4) be a square-free integer such that x 2 - d y 2 = - 1 and x 2 - d y 2 = 6 are solvable in integers. We prove the existence of infinitely many quadruples in Z [ d ] with the property D(n) when n ∈ { (4 m + 1) + 4 k d , (4 m + 1) + (4 k + 2) d , (4 m + 3) + 4 k d , (4 m + 3) + (4 k + 2) d , (4 m + 2) + (4 k + 2) d } for m , k ∈ Z . As a consequence, we provide few counter examples to Conjecture 1 of Franušić and Jadrijević [Math. Slovaca 69, 1263–1278 (2019)]. [ABSTRACT FROM AUTHOR]

We prove that for every integer n, there exist infinitely many D(n)-triples which are also D(t)-triples for t ∈ Z with n ≠ t . We also prove that there are infinitely many D (- 1) -triples in Z [ i ] which are also D(n)-triple in Z [ i ] for two distinct n's other than n = - 1 and these triples are not equivalent to any triple with the property D(1). [ABSTRACT FROM AUTHOR]

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form p = 8k +1, receives attention. [ABSTRACT FROM AUTHOR]

For a number field F and a prime number p, the ℤp-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T p (F) , is an important subject in abelian p-ramification theory. In this paper, we study the group T 2 (F) = T 2 (m) of the quadratic field F = ℚ (m) . Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk 4 (T 2 (− m)) = rk 2 (T 2 (− m)) − rank (R) , where R is a certain explicitly described Rédei matrix over F 2 . Furthermore, using this formula, we obtain the 4-rank density formula of T 2 -groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain results about the 2-power divisibility of orders of T 2 (± l) and T 2 (± 2 l) , both of which are cyclic 2-groups. In particular, we find that # T 2 (l) ≡ 2 # T 2 (2 l) ≡ h 2 (− 2 l) (mod 16) if l ≡ 7 (mod 8), where h2(−2l) is the 2-class number of ℚ (− 2 l) . We then obtain density results for T 2 (± l) and T 2 (± 2 l) when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about T p (F) when F varies over real or imaginary quadratic fields for any prime p, and about T 2 (± l) and T 2 (± 2 l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T 2 (l) case is closely connected to Shanks-Sime-Washington's speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units. [ABSTRACT FROM AUTHOR]

In a recent paper, Kishi and Komatsu have shown that the complex quadratic fields of discriminant Δ 18 = 4 - 3 18 n + 3 have a class group whose 3-Sylow subgroup is of rank at least three for all integers n. The Kishi and Komatsu paper proves this by looking at the structure of the algebraic fields involved. Our purpose in this paper is to show that the class groups for the order whose discriminant is Δ 18 have a 3-SSG of rank at least three, to show this using elementary means that would have been available to Gauss, and to elucidate further the group structure of the subgroup of classes of order 3. More generally, we shall show that Δ 18 is actually a special case of a more general sequence of discriminants, all of which can be shown parametrically to have a class group whose 3-Sylow subgroup has rank at least three. [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]

Let d be a square-free positive integer and CL (- d) the ideal class group of the imaginary quadratic field Q (- d) . In this paper, we show that given any odd integer g ≥ 3 and any integers r ≥ 1 , s with 0 ≤ s ≤ r , there are infinitely many d such that CL (- d) contains an element of order g and CL (- d) / CL (- d) 4 ≅ (Z / 2 Z) r - s × (Z / 4 Z) s . [ABSTRACT FROM AUTHOR]