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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]