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.
What is the purpose of the characteristic exponent?
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$\begingroup$
terminology
field-theory
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1On 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=false – 2012-09-04
1 Answers
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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.