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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]

This article is devoted to the even entire functions, which are the exact solutions for the Laplace and diffusion equations. These functions are considered in the algebraic number field. We guess that the functions have purely real zeros in the entire complex plane. These are proposed as new connections with algebraic number theory and mathematical physics. [ABSTRACT FROM AUTHOR]

Let n be a positive integer. We show that if the equation (1) x 4 + 2 n y 4 = z 4 has a solution (x,y,z) in a cubic number field K with x y z ≠ 0 , then the Galois group of the field K is the symmetric group S3. In addition, we show that for every positive integer d > 1 , there exists a number field Kd of degree d such that equation (1) has a solution (x,y,z) in Kd with x y z ≠ 0 . This paper extends the recent work of Bremner and Choudhry [5]. [ABSTRACT FROM AUTHOR]

In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let K be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in K in terms of the size of the input and the parameters of K. We include some earlier results about these, but we go further than them, e.g. we also analyze some R -specific operations in K like less-than comparison. In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from Z n to K n , and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to floating-point approximations). [ABSTRACT FROM AUTHOR]

In this paper we show that for eachk∈ ℕ there are infinitely many algebraic integers with normkand absolute normalized size smaller than 1. We also show that the lower bound (n+slog 2)/2 on the square of the absolute size ‖α‖ of an algebraic integer α of degreenwith exactlysreal conjugates over ℚ is best possible for each evens> 2. For this, for each pairs,k∈ ℕ, wheresis even, we construct algebraic integers α with exactlysreal conjugates and norm of modulusksatisfying deg α =nand ‖α‖2= (n+slog 2)/2 + logk+O(n−1) asn→ ∞ to. Finally, using the third smallest Pisot number θ3, which is the root of the polynomialx5–x4–x3+x2–1, we construct algebraic integers α of degreenthat have exactly one real conjugate and satisfy ‖α‖2≤ n/2 + 0.346981 … (which is quite close to the above lower bound (n+ log 2)/2 = n/2 + 0.346573 … fors= 1). In the proofs we use some irreducibility theorems for lacunary polynomials and the Erdős and Turán bound on the number of roots of a polynomial in a sector. [ABSTRACT FROM AUTHOR]

D. A. Goldston, J. Pintz and C. Y. Yıldırım [3] proved the existence of infinitely many prime pairs whose difference is arbitrarily small compared to the average gap, namely $$ \liminf_{n\to\infty} \frac{p_{n+1}-p_n}{\log p_n}=0. $$ In the present work we generalize their result to totally real number fields. We prove that if ω0 and ω1 run over distinct prime elements of the number field, then $$ \liminf_{\omega_0,\omega_1} \left|\frac{{\mathbf {N}}(\omega_1-\omega_0)}{{\mathbf {N}}\omega_0}\right|=0. $$ [ABSTRACT FROM AUTHOR]

Abstract: For an odd prime number p, we consider the p-primary part of the Brumer–Stark conjecture for a cyclic extension of number fields of degree 2p. We extend earlier work of Greither, Roblot, and Tangedal (2004) by proving the conjecture when the minus component of the p-primary part of the class group of K is not a cyclic Galois module. Consequently, we are able to prove the full Brumer–Stark conjecture for some new classes of number field extensions. [Copyright &y& Elsevier]

We prove the nonvanishing lemma for explicit second kind Padé approximations to generalized hypergeometric and q-hypergeometric functions. The proof is based on an evaluation of a generalized Vandermonde determinant. Also, some immediate applications to the Diophantine approximation is given in the form of sharp linear independence measures for hypergeometric E- and G-functions in algebraic number fields with different valuations. [ABSTRACT FROM AUTHOR]

Abstract: A practical computational system is described for computing with an algebraic closure of a field. The system avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. All roots of an arbitrary polynomial defined over such an algebraically closed field can be constructed and are easily distinguished within the system. The difficult case of inseparable extensions of function fields of positive characteristic is also handled properly by the system. A technique of modular evaluation into a finite field critically ensures that a unique genuine field is simulated by the system but also provides fast optimizations for some fundamental operations. Fast matrix techniques are also used for several non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation. [Copyright &y& Elsevier]

Abstract: Text: By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59–73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=c61xM-5D6Do. [Copyright &y& Elsevier]

Abstract: The groups in this paper are abelian. Let G be a reduced torsion-free finite rank group. Then G is cocommutative if is commutative modulo the nil radical. The class number of G, , is the number of isomorphism classes of groups H that are locally isomorphic ( isomorphic) to G. We say that G satisfies the power cancellation property if for some group H and integer implies that . We say that G has a Σ-unique decomposition if has a unique direct sum decomposition for each integer . Let for some group and some integer . We say that G has internal cancellation if given such that then . We use the class number to study the torsion-free finite rank groups G that have the power cancellation property, or a Σ-unique decompositions, or the internal cancellation property. Furthermore, we show that the power cancellation property for cocommutative strongly indecomposable reduced torsion-free finite rank groups is equivalent to the problem of determining the class number of an algebraic number field. Let be the integral closure of G. Using the Mayer–Vietoris sequence we show that there are finite groups associated with G of orders , , and such that . Let where p is a rational prime and is the ring of algebraic integers in the algebraic number field . Let be a group such that . We show that the sequence is asymptotically equal to the sequence where . Furthermore, for quadratic number fields k, iff is asymptotically equal to the sequence of rational primes. This connects unique factorization in number fields with the sequence of rational primes, and with direct sum properties of integrally closed cocommutative strongly indecomposable torsion-free finite rank groups. [Copyright &y& Elsevier]

This paper concerns the description of holomorphic extensions of algebraic number fields. After expanding the notion of adele class group to number fields of infinite degree over ℚ, a hyperbolized adele class group ${\hat{\mathfrak{S}}}_{K}$ is assigned to every number field K/ℚ. The projectivization of the Hardy space ℙ햧•[K] of graded-holomorphic functions on ${\hat{\mathfrak{S}}}_{K}$ possesses two operations ⊕ and ⊗ giving it the structure of a nonlinear field extension of K. We show that the Galois theory of these nonlinear number fields coincides with their discrete counterparts in that 햦햺헅(ℙ햧•[K]/K) = 1 and 햦햺헅(ℙ햧•[L]/ℙ햧•[K]) ≅ 햦햺헅(L/K) if L/K is Galois. If Kab denotes the maximal abelian extension of K and 햢K is the idele class group, it is shown that there are embeddings of 햢K into 햦햺헅⊕(ℙ햧•[Kab]/K) and 햦햺헅⊗(ℙ햧•[Kab]/K), the "Galois groups" of automorphisms preserving ⊕ (respectively, ⊗) only. [ABSTRACT FROM AUTHOR]

In this paper, we prove that some power series with rational coefficients take either values of rational numbers or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. We then apply this result to an algebraic number field. In addition, we establish the p-adic analogues of these results and show that these results have analogues in the field of p-adic numbers. [ABSTRACT FROM AUTHOR]