### A differential criterion for complete intersections.

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We extend the theory of spinor class fields and relative spinor class fields to study representation problems in several classical linear algebraic groups over number fields. We apply this theory to study the set of isomorphism classes of maximal orders of central simple algebras containing a given maximal Abelian suborder. We also study isometric embeddings of one skew-Hermitian Quaternionic lattice into another.

In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field...

Let $R$ be a non-maximal order in a finite algebraic number field with integral closure $\overline{R}$. Although $R$ is not a unique factorization domain, we obtain a positive integer $N$ and a family $\mathcal{Q}$ (called a Cale basis) of primary irreducible elements of $R$ such that ${x}^{N}$ has a unique factorization into elements of $\mathcal{Q}$ for each $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm{Spec}\left(\overline{R}\right)\to \mathrm{Spec}\left(R\right)$ is bijective. This last condition is actually equivalent to several properties linked...

The question of embedding fields into central simple algebras $B$ over a number field $K$ was the realm of class field theory. The subject of embedding orders contained in the ring of integers of maximal subfields $L$ of such an algebra into orders in that algebra is more nuanced. The first such result along those lines is an elegant result of Chevalley [6] which says that with $B={M}_{n}\left(K\right)$ the ratio of the number of isomorphism classes of maximal orders in $B$ into which the ring of integers of $L$ can be embedded...

Let ε be an algebraic unit for which the rank of the group of units of the order ℤ[ε] is equal to 1. Assume that ε is not a complex root of unity. It is natural to wonder whether ε is a fundamental unit of this order. It turns out that the answer is in general yes, and that a fundamental unit of this order can be explicitly given (as an explicit polynomial in ε) in the rare cases when the answer is no. This paper is a self-contained exposition of the solution to this problem, solution which was...