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This question is related to this answer. I always thought that we can simply say since $2 \in \mathbb{Z}$, and $2^{-1} = 1/2 \notin \mathbb{Z}$, $\mathbb{Z}$ is not a field.

Can someone explain why this proof is wrong as the answer claims?

Community
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Dominic
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  • This just seems like a duplicate of that question. Once you gain 50 reputation points on the site, you will be able to comment there to ask for clarification from the person who wrote that answer. – Zev Chonoles May 30 '13 at 22:52
  • The rational number $1/2$? – Dominic May 30 '13 at 22:53
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    My point was, how would you define $\frac{1}{2}$? But this is really what the original thread is addressing. – Zev Chonoles May 30 '13 at 22:56
  • It depends on the level of formality of the course. For most purposes, as in giving an example of non-field, I would consider roughly what you wrote enough, though I would prefer "it is clear that there is no integer $x$ such that $2x=1$." (That stays in the integers, does not appeal to external entities.) – André Nicolas May 30 '13 at 23:03

1 Answers1

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Two is not equal to zero. Two times an integer is an even integer. Since $1$ is odd, there is no integer $n$ with $2n = 1$.

Jay
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