# The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)

## Problem 519

Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).

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## Proof.

Every element of the ring $\Z[\sqrt{5}]$ can be written as $a+b\sqrt{5}$ for some integers $a, b$.

The (field) norm $N$ of an element $a+b\sqrt{5}$ is defined by

\[N(a+b\sqrt{5})=(a+b\sqrt{5})(a-b\sqrt{5})=a^2-5b^2.\]

Consider the case when $a=3, b=1$.

Then we have

\[(3+\sqrt{5})(3-\sqrt{5})=4=2\cdot 2. \tag{*}\]

We prove that elements $2, 3\pm \sqrt{5}$ are irreducible in $\Z[\sqrt{5}]$.

Note that the norms of these elements are $4$.

**We claim that each element $\alpha \in \Z[\sqrt{5}]$ of norm $4$ is irreducible**.

Suppose that $\alpha=\beta \gamma$ for some $\beta, \gamma \in \Z[\sqrt{5}]$.

Our objective is to show that either $\beta$ or $\gamma$ is a unit.

Since we have

\[4=N(\alpha)=N(\beta \gamma)=N(\beta) N(\gamma)\]
and the norms are integers, the value of $N(\beta)$ is one of $\pm 1, \pm 2, \pm 4$.

If $N(\beta)=\pm 1$, then $\beta$ is a unit.

If $N(\beta)=\pm 4$, then $N(\gamma)=\pm 1$ and hence $\gamma$ is a unit.

Let us consider the case $N(\beta)=\pm 2$.

We show that this case does not happen.

Write $\beta=a+b\sqrt{5}$ for some integers $a, b$.

Then we have

\[\pm 2 =N(\beta)=a^2-5b^2.\]
Considering the above equality modulo $5$ yields that

\[\pm 2 \equiv a^2 \pmod{5}.\]
However note that any square of an integer modulo $5$ is one of $0, 1, 4$.

So this shows that there is no such $a$.

Therefore, we have proved that either $\beta$ or $\gamma$ is a unit, hence $\alpha$ is irreducible.

The claim is proved.

It follows from (*) that the element $4 \in \Z[\sqrt{5}]$ has two different decompositions into irreducible elements.

Thus the ring $\Z[\sqrt{5}]$ is not a UFD.

## Related Question.

**Problem**.

Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD).

This problem only differs from the current problem by the sign.

($-5$ is used instead of $5$.)

For a proof of this problem, check out the post ↴

The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD)

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