Let $R$ be a field and consider $$A=\{n\in \mathbb N^*\mid \underbrace{1+...+1}_{n\ times}=0\}.$$ Assuming that it's not empty, prove that its smallest element is prime.

I have no idea how to do it. It looks here that we are in $\mathbb Z/n\mathbb Z$ and since $R$ is a field and that $A$ might be a subfield, then $n$ will be prime, but this don't answer the question since they ask for the smallest element.

If my question is unclear, let me know.

  • $\begingroup$ $A$ cannot be a subfield because it is a set of integers, not a subset of the field at all. It is almost an ideal of $\mathbb Z$, except that it lacks the non-positive elements. $\endgroup$ – hmakholm left over Monica Sep 25 '16 at 12:12
  • $\begingroup$ By the way, what is $\mathbb N^*$? $\endgroup$ – hmakholm left over Monica Sep 25 '16 at 12:13
  • $\begingroup$ @HenningMakholm: What could it be except $\mathbb N\backslash \{0\}$ ? $\endgroup$ – user352653 Sep 25 '16 at 12:42
  • $\begingroup$ See here. $\endgroup$ – user228113 Sep 25 '16 at 13:08

Assume its smallest element $n$ is not prime and write $n=ab$ where $a\geq 2$, $b\geq 2$. Then: $$\underbrace{(\underbrace{1+...+1}_{a\ times})+(\underbrace{1+...+1}_{a\ times})+\cdots+(\underbrace{1+...+1}_{a\ times})}_{b\ times}=0$$ Rewrite this as $$b\cdot(\underbrace{1+...+1}_{a\ times})=0.$$ Can you spot a contradiction?

  • $\begingroup$ Either $b=0$ or $a=0$ ? which contradict $a,b\geq 2$ ? But don't we directly have that $ab=n=0$ which is a contradiction ? $\endgroup$ – user352653 Sep 25 '16 at 12:34
  • $\begingroup$ but $b$ is not an element of $R$ $\endgroup$ – mercio Sep 25 '16 at 13:39
  • $\begingroup$ $b$ is shorthand for $b\cdot 1$ which is well defined. $\endgroup$ – Olivier Moschetta Sep 25 '16 at 13:46
  • $\begingroup$ Actually, writing it as $$(\underbrace{1+\cdots+1}_{b\text{ times}})\cdot(\underbrace{1+\cdots+1}_{a\text{ times}})=0$$ would be clearer. - @user352653: The originally assumed $n=ab$ is an equation in the integers, not in $R$, so $n\ne 0$ by assumption. $\endgroup$ – hmakholm left over Monica Sep 25 '16 at 14:09

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