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Which of the following integral domains are Euclidean domains?

  1. $\mathbb{Z}[\sqrt{-3}]$
  2. $\mathbb{Z}[x]$
  3. $\mathbb{R}[x^2,x^3]=\{f=\sum_{i=0}^n a_ix^i\in\mathbb{R}[x]:a_1=0\}$
  4. $(\mathbb{Z}[x]/(2,x))[y]$

How can we solve this problem. Can anyone suggest me something. Thanks

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To show that something is a ED, find a Euclidean function. To show thta something isn't, e.g show that it's not a PID. – Alex B. Dec 17 '12 at 9:12
@priti: Something is very strange in you definition of (3) – Dennis Gulko Dec 17 '12 at 9:26
@AlexB., ok if it works, but there are PIDs that are not Euclidean: – lhf Dec 17 '12 at 10:35

Note that every ED is PID and every PID is UFD. Now comming to your question,

  • $\mathbb{Z}[\sqrt{3}i]$ is not UFD because $(1+\sqrt{3}i)(1-\sqrt{3}i)=4=2.2$
  • $\mathbb{Z}[x] $ is not PID because $(2,x)$ is not principal.
  • Note that $x^2$ and $x^3$ are irreducible in $\mathbb{R}[x^2,x^3]$ and $(x^2)^3=x^6=(x^3)^2,$ hence it is not a UFD.
  • $\mathbb{Z}[x]/(2,x)\cong\mathbb{F}_2$ which is a field, hence $\mathbb{F}_2[y]$ is ED.

Hence the correct option is the $4$th one.

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