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Let $P=\{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$.

a) Which elements are irreducible in $P$: 4, 5, 6, 9, 10, 15?

b) Find out, which one of rings: $ P$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ is a unique factorization domain.

My guess for a) is 4 and 10. Could you give me any hints for b)?

user26857
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MatJ
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1 Answers1

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a) $4=\frac{2}{1}\cdot\frac21$ and similarly for $10$. On the other side, $5$ is irreducible, $6=\frac21\cdot\frac31$ and $\frac31$ is invertible in $P$, so $6$ is also irreducible, and similarly for $15$. It remains $9=3^2$ which is invertible in $P$.

b) $P$ is a UFD as a ring of fractions of a UFD; see here. We also have that $P[x]$ is a UFD; see here.

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user26857
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  • $\mathbb Z[\sqrt{-5}]$ is well known as not being a UFD for this simple reason: $2\cdot 3=(1+\sqrt{-5})(1-\sqrt{-5})$. – user26857 Dec 17 '14 at 19:39