A primitive element theorem for fields with commuting derivations and automorphisms
A primitive element theorem for fields with commuting derivations and automorphisms
复制标题
具有交换导数和自同构的域的本原元定理
DOI：
10.1007/s00029-019-0504-9
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Pogudin, Gleb
中科院分区：
文献类型：
--
作者：
Pogudin, Gleb
我们建立了一个域的本原元定理，该域具有多个交换算子，使得每个算子要么是导子，要么是自同构。更精确地说，我们证明了：对于这样的零特征域的每一个扩张，使得E是由F上的n个元素用域运算和算子生成的，E的每一个元素都满足一个系数在F中的包含域运算和算子的非平凡方程，算子对E的作用是无余的，存在一个元素使得E是由F上的域运算和算子生成的.这个结果同时在两个方向上推广了Kolchin和Cohn的本原元定理：我们允许任意数目的导子和自同构，并且对基场F不施加任何限制。
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 extensionof such fields of zero characteristic such thatEis generated overFby finitely many elements using the field operations and the operators,every element ofEsatisfies a nontrivial equation with coefficient inFinvolving the field operations and the operators,the action of the operators onEis irredundantthere exists an elementsuch thatEis generated overFbyausing 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 fieldF.