# What is the purpose of the characteristic exponent?

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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Did you find the term only on wikipedia or also somewhere else? I would bet that it is related to the last possible meaning on the disambiguity page: "An exponent of a power series with non-negative coefficient, that is not divisible by the highest common factor of preceding exponents with non-zero coefficients." – Simon Markett Sep 4 '12 at 17:12
No, it's from a paper, it is about fields and it is defined there exactly the way I defined it. – Jesko Hüttenhain Sep 4 '12 at 17:20
On first glance, it looks like a convenient way to shorten hypotheses: books.google.ca/… – Andrew Sep 4 '12 at 17:23

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.