2015/09/29 by H. W. Lenstra, Alice Silverberg, Lenstra, H. W. +1
Computer Science · Engineering · Mathematics · #68W30 #Algebraic structures and combinatorial models #Coding theory and cryptography #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Primary: 13E10 #Secondary: 13P99 #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1509.08843
openalex publication_date 2015/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel-Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.