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I just came across the term "characteristic exponent" of a field $\Bbbk$. Apparently, it is equal to $1$ if $\DeclareMathOperator{\c}{char}\c(\Bbbk)=0$ and it is equal to $p=\c(\Bbbk)$ otherwise. Going to wikipedia was extremely unhelpful as you can see when searching for the term "characteristic exponent" on the page it redirects you to.

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    On first glance, it looks like a convenient way to shorten hypotheses: http://books.google.ca/books?id=bDLecF34d8UC&pg=PA113&lpg=PA113&dq=field+characteristic+exponent&source=bl&ots=b_uV5JqeDb&sig=Vqcqov3ARy9j3n4aL1zNsjOF52k&hl=en#v=onepage&q=field%20characteristic%20exponent&f=false2012-09-04

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It simplifies some statements of theorems and properties. For instance one could define

A field $K$ of characteristic exponent $p$ is perfect, if $K^p = K$.

Or, suppose you have a field $K$ of characteristic exponent $p$ and you want to study its connection to $\mathbb{Z}[1/p]$ or to some $\mathbb{Z}[1/p]$-module. It would be tedious to consider $\mbox{char}(K)=0$ seperately.