The title is from the first paragraph of Serre's 'A Course in Arithmetic.' What does $n.1$ = 0 mean? I know $p$ is the smallest positive integer to be congruent to 0, but what does $n.1$ mean?
For a field $K,$ if char($K$) = $p, p$ is the smallest integer $n$ > 0 such that $n.1 = 0. $
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number-theory
elementary-number-theory
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0$n. 1=\overbrace{1+\cdots+1}^{n\,times}$ – 2017-02-15
1 Answers
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The notation $n.1$ means that $1+\ldots+1$ $n$-times. It is the image of $n$ by the unique ring homomorphism from $\mathbb{Z}$ to $K$.
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0Is there a reason why we don't just call that $n$? Is it because $n$ could be mapped to its residue class mod $p$ so it isn't really $n?$ – 2017-02-15
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1It is to avoid confusion, $n$ is an integer, while $n.1$ is an element of $K$. Notice that $\mathbb{Z}\hookrightarrow K$ if and only if $\textrm{char}(K)=0$. So the only situation where it is reasonable to write $n$ for $n.1$ is when $\textrm{char}(K)=0$. Besides, you seem to assume that $\mathbb{Z}/p\mathbb{Z}$ is the only field of characteristic $p$. Which is not true, consider $$(\mathbb{Z}/2\mathbb{Z}[x])/(x^2+x+1).$$ – 2017-02-15