2018/12/29 by Gleb Pogudin, Pogudin, Gleb
Mathematics · #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.DS #math.RA
paper · pdf · doi:10.48550/arxiv.1812.11375
openalex publication_date 2018/12/29 · arxiv created 2019/09/13 · arxiv updated 2019/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension F ⊂ E of such fields of zero characteristic such that \bullet E is generated over F by finitely many elements using the field operations and the operators, \bullet every element of E satisfies a nontrivial equation with coefficient in F involving the field operations and the operators, \bullet the action of the operators on E is irredundant there exists an element a ∈ E such that E is generated over F by a using the field operations and the operators. This result generalizes the Primitive Element Theorems by Kolchin and Cohn in two directions simultaneously: we allow any numbers of derivations and automorphisms and do not impose any restrictions on the base field F.