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Let p ≡ 1 (mod 8) and q ≡ 7 (mod 8) be two prime numbers. The purpose of this paper is to compute the unit groups of the fields L = Q (2 , p , q) and give their 2-class numbers. [ABSTRACT FROM AUTHOR]

For an infinite family of real biquadratic fields k we give the structure of the Iwasawa module X = X (k ∞) of the Z 2 -extension of k. For these fields, we obtain that λ = μ = 0 and ν = 2 . where λ , μ and ν are the Iwasawa invariants of the cyclotomic Z 2 -extension of k [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]

We state and prove a Gras-type equality relating a certain module generated by Rubin–Stark units and the ray class number. We use for this purpose the notion of generalized index. [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 note, we prove that for every even integer d ≥ 4 , there is a number field K of degree d over Q , such that for each sufficiently large N ∈ N among (2 N + 1) d distinct elements of the form a 1 ω 1 + ⋯ + a d ω d , where ω 1 , ⋯ , ω d is an integral basis of the ring O K and a 1 , ⋯ , a d ∈ Z , | a 1 | , ⋯ , | a d | ≤ N , at least c (log N) d / 2 - 1 elements are unimodular. Here, c is a positive constant that depends on the field and the integral basis only. In particular, for a quartic field K, the lower bound for the number of unimodular elements of such form is at least c log N . We show that that this estimate is best possible for any quartic field K and any ω 1 , ⋯ , ω 4 ∈ O K . [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group and Γ ⊂ End (G) be a subset of the endomorphism group of G. A sequence S = g 1 · ... · g ℓ over G is a (Γ -)weighted zero-sum sequence if there are γ 1 , ... , γ ℓ ∈ Γ such that γ 1 (g 1) + ... + γ ℓ (g ℓ) = 0 . We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group Γ ) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences. [ABSTRACT FROM AUTHOR]

The arithmetic nature of the Euler's constant γ is one of the biggest unsolved problems in number theory from almost three centuries. In an attempt to give a partial answer to the arithmetic nature of γ , Murty and Saradha made a conjecture on linear independence of digamma values. In particular, they conjectured that for any positive integer q > 1 and a field K over which the q-th cyclotomic polynomial is irreducible, the digamma values namely ψ (a / q) where 1 ≤ a ≤ q with (a , q) = 1 are linearly independent over K. Further, they established a connection between the arithmetic nature of the Euler's constant γ to the above conjecture. In this article, we first prove that the conjecture is true with at most one exceptional q. Later on we also make some remarks on the linear independence of these digamma values with the arithmetic nature of the Euler's constant γ . [ABSTRACT FROM AUTHOR]

For an imaginary biquadratic number field L = Q (i , d) , where d is an odd square-free integer, let L ∞ be the cyclotomic Z 2 -extension of L. For any integer n ≥ 0 , we denote by Ln the nth layer of L ∞ / L . We study the rank of the 2-primary part of the class group of Ln and then we draw the list of all number fields L where the Galois group of the maximal unramified pro-2-extension of L ∞ is metacyclic. [ABSTRACT FROM AUTHOR]

For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x ∈ M {x\in M} , denoted LD ⁡ (x) {\operatorname{LD}(x)} , and the entire monoid M, denoted LD ⁡ (M) {\operatorname{LD}(M)}. This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L ⁢ (x) {L(x)} , ℓ ⁢ (x) {\ell(x)} , and ρ ⁢ (x) {\rho(x)}. We consider some general properties of LD ⁡ (x) {\operatorname{LD}(x)} and LD ⁡ (M) {\operatorname{LD}(M)} and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD ⁡ (M) {\operatorname{LD}(M)} is rational and there is a nonunit element x ∈ M {x\in M} with LD ⁡ (M) = LD ⁡ (x) {\operatorname{LD}(M)=\operatorname{LD}(x)} (such a monoid is said to have accepted length density). While it is well known that the much studied asymptotic versions of L ⁢ (x) {L(x)} , ℓ ⁢ (x) {\ell(x)} , and ρ ⁢ (x) {\rho(x)} (denoted L ¯ ⁢ (x) {\overline{L}(x)} , ℓ ¯ ⁢ (x) {\overline{\ell}(x)} , and ρ ¯ ⁢ (x) {\overline{\rho}(x)}) always exist, we show the somewhat surprising result that LD ¯ ⁢ (x) = lim n → ∞ ⁡ LD ⁡ (x n) {\overline{\operatorname{LD}}(x)=\lim_{n\rightarrow\infty}\operatorname{LD}(x^% {n})} may not exist. We also give some finiteness conditions on M that force the existence of LD ¯ ⁢ (x) {\overline{\operatorname{LD}}(x)}. [ABSTRACT FROM AUTHOR]

The modular trace of the normalized Hauptmodul has been extended to the Galois trace of a class invariant by Kaneko. It is an important issue to search for class invariants for which the Galois traces have modular properties. The crucial point in this paper is that we initiate a new notion, called Siegel resolvents; we define the Siegel resolvents as the quadratic polynomials of Siegel functions of level 3, so that they are modular functions of level 3 as well. We construct real-valued class invariants over imaginary quadratic fields by using the singular values of Siegel resolvents at imaginary quadratic irrationals. We also prove that the generating series of their Galois traces become a weakly holomorphic modular form with weight 3/2. This shows that the work of D. Zagier on traces of singular moduli can be extended to the modular functions of higher level. [ABSTRACT FROM AUTHOR]

Let d be an odd square-free integer and ζ 8 a primitive 8-th root of unity. The purpose of this paper is to investigate the rank of the 2-class group of the fields L d = Q (ζ 8 , d) . [ABSTRACT FROM AUTHOR]

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one.

In this paper we show if E is a totally real Galois extension of Q of degree s then there is a cubic extension K of E with signature (s, s) such that in the group of units of K there are s multiplicatively independent positive units with the following property: if α is any of those s units then there are two conjugates of α over Q which are complex conjugate numbers lying on the circle | z | = α - 1 / 2 . In particular, our result combined with a recent result of Otiman implies that the corresponding Oeljeklaus–Toma manifold X(K, U) admits a pluriclosed metric. [ABSTRACT FROM AUTHOR]

Let be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and its second Hilbert 2-class field. Denote by the Galois group of . The purpose of this note is to investigate the Hilbert 2-class field tower of and then deduce the structure of G. [ABSTRACT FROM AUTHOR]

After Zagier's significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura's reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level N ≥ 2 and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber's resolvents, under revision). [ABSTRACT FROM AUTHOR]

Let H be a transfer Krull monoid over a subset G0 of an abelian group G with finite exponent. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · ... · u k. The set L (a) of all possible factorization lengths k is called the set of lengths of a, and H is said to be half-factorial if | L (a) | = 1 for all a ∈ H. We show that, if a ∈ H is a non-unit and | L (a ⌊ (3 exp (G) − 3) / 2 ⌋ ) | = 1 , then the smallest divisor-closed submonoid of H containing a is half-factorial. In addition, we prove that, if G0 is finite and | L (∏ g ∈ G 0 g 2 ord (g) ) | = 1 , then H is half-factorial. [ABSTRACT FROM AUTHOR]

Let ε be an algebraic unit of the degree n ⩾ 3. Assume that the extension ℚ(ε)/ℚ is Galois. We would like to determine when the order ℤ[ε] of ℚ(ε) is Gal(ℚ(ε)/ℚ)-invariant, i.e. when the n complex conjugates ε1, ..., εn of ε are in ℤ[ε], which amounts to asking that ℤ[ε1, ..., εn] = ℤ[ε], i.e., that these two orders of ℚ(ε) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order ℤ[ε1, ε2, ε3]. However, there is no known similar formula for n > 3. In the present paper, we put forward and motivate three conjectures for the solution to this determination for n = 4 (two possible Galois groups) and n = 5 (one possible Galois group). In particular, we conjecture that there are only finitely many cyclic quartic and quintic Galois-invariant orders generated by an algebraic unit. As a consequence of our work, we found a parametrized family of monic quartic polynomials in ℤ[X] whose roots ε generate bicyclic biquadratic extensions ℚ(ε)/ℚ for which the order ℤ[ε] is Gal(ℚ(ε)/ℚ)-invariant and for which a system of fundamental units of ℤ[ε] is known. According to the present work it should be difficult to find other similar families than this one and the family of the simplest cubic fields. [ABSTRACT FROM AUTHOR]

In this article, we show that certain abelian extensions K with unit rank greater than or equal to three have cyclic class group if and only if it has a Euclidean ideal class. This result improves an earlier result of Murty and Graves. One can improve this result up to unit rank 2 if one assumes the Elliott and Halberstam conjecture (see Conjecture 1 in preliminaries). These results are known under generalized Riemann hypothesis by the work of Lenstra (J Lond Math Soc 10:457–465) [see also Weinberger (Proc Symp Pure Math 24:321–332)]. [ABSTRACT FROM AUTHOR]

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal. [ABSTRACT FROM AUTHOR]

Let d be a non-square positive integer such that the period of the continued fraction expansion of d is even. We give some relations between some properties of partial quotients of the continued fraction expansion of d , which emerge from numerical experiments. [ABSTRACT FROM AUTHOR]

Let K be a number field. In the terminology of Nagell a unit ε of K is called exceptional if 1 − ε is also a unit. The existence of such a unit is equivalent to the fact that the unit equation ε 1 + ε 2 + ε 3 = 0 is solvable in units ε 1 , ε 2 , ε 3 of K . Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer k with k ≥ 3 , denoted by ℓ ( K ) , such that the unit equation ε 1 + … + ε k = 0 is solvable in units ε 1 , … , ε k of K . If no such k exists, we set ℓ ( K ) = ∞ . Apart from trivial cases when ℓ ( K ) = ∞ , we give an explicit upper bound for ℓ ( K ) . We obtain several results for ℓ ( K ) in number fields of degree at most 4, cyclotomic fields and general number fields of given degree. We prove various properties of ℓ ( K ) , including its magnitude, parity as well as the cardinality of number fields K with given degree and given odd resp. even value ℓ ( K ) . Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions. [ABSTRACT FROM AUTHOR]

Abstract: We investigate in detail a homomorphism which we call the 2‐Selmer signature map from the 2‐Selmer group of a number field K to a nondegenerate symmetric space, in particular proving the image is a maximal totally isotropic subspace. Applications include precise predictions on the density of fields K with given narrow class group 2‐rank and with given unit group signature rank. In addition to theoretical evidence, extensive computations for totally real cubic and quintic fields are presented that match the predictions extremely well. In an Appendix with Richard Foote, we classify the maximal totally isotropic subspaces of orthogonal direct sums of two nondegenerate symmetric spaces over perfect fields of characteristic 2 and derive some consequences, including a mass formula for such subspaces. [ABSTRACT FROM AUTHOR]