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Problem 13 (Gaussian Integers Under Modular Arithmetic)
Consider the commutative rings $\mathbb{Z}_p[i]=\{a+bi \mid a,b\in \mathbb{Z}_p \}$ for $p\in\{2,3,5\}$. In this problem we'll analyze the multiplicative structure and determine which ring are integral domains, and which are fields.
- Construct a multiplication table for $\mathbb{Z}_2[i]$. Is this ring an integral domain? Is it a field?
- Construct a multiplication table for $\mathbb{Z}_3[i]$. Is this ring an integral domain? Is it a field?
- Is $\mathbb{Z}_5[i]$ an integral domain? We'll soon show that any finite integral domain is a field.
- Optional: Make a conjecture for which $p$ the ring $\mathbb{Z}_p[i]=\{a+bi \mid a,b\in \mathbb{Z}_p \}$ is an integral domain.
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