# A note on the kernels of higher derivations

• Volume: 63, Issue: 3, page 583-588
• ISSN: 0011-4642

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## Abstract

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Let $k\subseteq {k}^{\text{'}}$ be a field extension. We give relations between the kernels of higher derivations on $k\left[X\right]$ and ${k}^{\text{'}}\left[X\right]$, where $k\left[X\right]:=k\left[{x}_{1},\cdots ,{x}_{n}\right]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D={\left\{{D}_{n}\right\}}_{n=0}^{\infty }$ a higher $k$-derivation on $k\left[X\right]$ and ${D}^{\text{'}}={\left\{{D}_{n}^{\text{'}}\right\}}_{n=0}^{\infty }$ a higher ${k}^{\text{'}}$-derivation on ${k}^{\text{'}}\left[X\right]$ such that ${D}_{m}^{\text{'}}\left({x}_{i}\right)={D}_{m}\left({x}_{i}\right)$ for all $m\ge 0$ and $i=1,2,\cdots ,n$. Then (1) $k{\left[X\right]}^{D}=k$ if and only if ${k}^{\text{'}}{\left[X\right]}^{{D}^{\text{'}}}={k}^{\text{'}}$; (2) $k{\left[X\right]}^{D}$ is a finitely generated $k$-algebra if and only if ${k}^{\text{'}}{\left[X\right]}^{{D}^{\text{'}}}$ is a finitely generated ${k}^{\text{'}}$-algebra. Furthermore, we also show that the kernel $k{\left[X\right]}^{D}$ of a higher derivation $D$ of $k\left[X\right]$ can be generated by a set of closed polynomials.

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