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**Drexel28** Yeah, it's hard to remember details sometime. $\displaystyle \mathbb{Z}_6$ and $\displaystyle \mathbb{Z}_{10}$ are not fields, the only quotient rings of $\displaystyle \mathbb{Z}$ which are fields are $\displaystyle \mathbb{Z}_p$ where $\displaystyle p$ is prime. And, no, the characteristic of a unital ring is it's order in the additive group. So, for example $\displaystyle \text{char}(\mathbb{Z}_6)=|1|_{\mathbb{Z}_6}=6$ and $\displaystyle \text{char}(\mathbb{Z}_{10})=|1|_{\mathbb{Z}_{10}} =10$.