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Let L = L1 ∪ L2 ∪ ⋯ ∪ Ln be a link in ℝ3 such that Li is a trivial link for each i, 1 ≤ i ≤ n. Let P1, P2,...,Pn be mutually distinct flat planes in ℝ3 such that no two of them are parallel. Then there is a link in ℝ3 such that L is ambient isotopic to L′ and for each i, 1 ≤ i ≤ n. [ABSTRACT FROM AUTHOR]

In this paper, we study the notion of norm-Euclidean ideal class, which was defined by Lenstra [Euclidean ideal classes, Astérisque 61 (1979) 121-131]. Using a slight modification of an algorithm determining among other properties the Euclidean minimum described in [P. Lezowski, Computation of the Euclidean minimum of algebraic number fields, preprint (2011); http://hal.archives-ouvertes.fr/hal-00632997/en/], we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari [The Euclidean condition in pure cubic and complex quartic fields, Math. Comput. 33 (1979) 389-398], we also establish the complete list of pure cubic number fields which admit a norm-Euclidean ideal class. Finally, we show that ${\mathbf{Q}}(\sqrt{2}, \sqrt{35})$, which is known to admit a non-principal Euclidean ideal class, has no norm-Euclidean ideal class. [ABSTRACT FROM AUTHOR]

In this paper, we give necessary and sufficient conditions for 8-ranks of narrow class groups of $F = \mathbb{Q}(\sqrt{2pq})$, distinct primes p ≡ q ≡ 1 mod 4. The results are useful for numerical computations. [ABSTRACT FROM AUTHOR]