Question? Is $Z$$_1$$_8$ a ring? Field?
This is only my second theoretical math class so I'm struggling heavily. I know for it to be a ring it needs to be 1). closed under addition/multiiplication, 2).assosciative laws must hold for add.multi, 3). distributivce laws must hold for add/multi, 4). commutative laws must hold for add./multi 5). additive identity, 6). additive inverse
My professor hasn't done many examples so I'm stuck on how to show the proof for each step.
1).Would this work? Let $\overline{a}$$\overline{b}$$\in$ $Z$$_1$$_8$, $\overline{a+ b}$$\in$ $Z$$_1$$_8$ b.c $m+n$ $\in$ $Z$ and same with multiplication?
With 2), 3), 4), wouldn't they all hold true because the set of all integers holds true?
First of all, for any $n\in \mathbb{N}$, $\mathbb{Z}_n$ is a commutative ring with unit. You can check every axiom by hand.
In general you can show that $\mathbb{Z}_n$ is a field if and only if $n$ is prime. This follows directly from the following property:
To prove the above property it is useful to remember Bézout's theorem.