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Problem 15 (Some Subrings Of The Complex Numbers)

Let $k$ be an integer. The sets $\mathbb{Z}[\sqrt{k}] = \{a+b\sqrt{k}\mid a,b\in \mathbb{Z}\} $ and $\mathbb{Q}[\sqrt{k}] = \{a+b\sqrt{k}\mid a,b\in \mathbb{Q}\} $ are clearly nonempty subsets of the field $\mathbb{C}$.

  1. Prove that both of these subsets are subrings of the complex numbers.
  2. Prove that $\mathbb{Z}[\sqrt{k}]$ is an integral domain.
  3. Prove that $\mathbb{Q}[\sqrt{k}]$ is a field.

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