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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]