#### Vol. 15, No. 1, 1965

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A new measure of a partial differential field extension

### Israel Zuckerman

Vol. 15 (1965), No. 1, 357–371
##### Abstract

Let $G$ be a differential field of characteristic zero with the commuting derivations ${d}_{1},\cdots \phantom{\rule{0.3em}{0ex}},{d}_{m}$. If $F$ is a differential subfield of $G$, the algebraic and differential degrees of transcendence of $G$ over $F$, denoted respectively by $d\left(G∕F\right)$ and $d.d\left(G∕F\right)$ are numerical invariants of the extension. Unlike the ordinary differential case $\left(m=1\right)$ $d.d.\left(G∕F\right)=0$ does not imply that $d\left(G∕F\right)$ is finite. In this paper an intermediate measure of the extension is constructed, called the limit vector. The first and last components of this vector correspond to $d.d\left(G∕F\right)$ and $d\left(G∕F\right)$ respectively, and the limit vector is additive.

Primary: 12.80