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Problem 2 (Algebraic Properties Of Modular Arithmetic)
Consider the sets $\mathbb{Z}_n$ for each positive integer $n$, together with modular addition and multiplication.
- Give three different integers $n$ so that $\mathbb{Z}_n$ is a field.
- Give an integer $n$ where $\mathbb{Z}_n$ is not a field. List the properties of being a field that are not satisfied.
- Find an integer $n$ and elements $a,b\in \mathbb{Z}_n$ with $a\neq b$ such that $ab=0$ but neither $a$ nor $b$ is zero.
- For each integer $k\geq 2$, find an integer $n$ and element $a\in \mathbb{Z}_n$ so that $a^k=0$ but $a^{k-1}\neq 0$.
- State all $n$ for which $\mathbb{Z}_n$ is a field.
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