# How to factorize $5$ in $\mathbb{Z}$?

Since $5$ has a norm of $125$ in this domain, and $N(1 + (\root 3 \of 2)^2) = 5$, it seems like a sensible proposition that $5 = (1 + (\root 3 \of 2)^2) \pi_2 \pi_3$, where $\pi_2, \pi_3$ are two other numbers in this domain having norms of $5$ or $-5$. This is supposed to be a unique factorization domain, right?

I am encouraged by the fact that $$N\left(\frac{5}{1 + (\root 3 \of 2)^2}\right) = N(1 + 2 \root 3 \of 2 – (\root 3 \of 2)^2) = 25.$$

But I am discouraged by the fact that $$\frac{5}{(-1 – (\root 3 \of 2)^2)(1 + (\root 3 \of 2)^2)} = \frac{7 – 6 \root 3 \of 2 – 2 (\root 3 \of 2)^2}{5}$$ is an algebraic number but not an algebraic integer. I have found a couple of other numbers with norms of $5$ or $-5$ but can’t get them to multiply to $5$ in any of the combinations of three of them that I have tried. I feel like I’m going around in circles.

#### Solutions Collecting From Web of "How to factorize $5$ in $\mathbb{Z}$?"

We have

$$\mathbf Z[\sqrt[3]{2}]/(5) \cong \mathbf Z[x]/(x^3 – 2, 5) \cong \mathbf Z_5[x]/(x^3 – 2) \cong \mathbf Z_5[x]/(x+2) \times \mathbf Z_5[x]/(x^2 + 3x + 4)$$

so that the ideal $(5)$ factors as $(5) = \mathfrak p_1 \mathfrak p_2$. To find the ideals $\mathfrak p_1$ and $\mathfrak p_2$, note that they correspond to the maximal ideals in the ring $\mathbf Z_5[x]/(x^3 – 2)$. We therefore have the following factorization:

$$(5) = (\sqrt[3]{2} + 2, 5)(\sqrt[3]{4} + 3 \sqrt[3]{2} + 4, 5)$$

Since the ring $\mathbf Z[\sqrt[3]{2}]$ is a principal ideal domain, these ideals are principally generated by prime elements of norm $5$ and $25$, respectively. Some easy trial and error by hand with the norm form yields that the first ideal is generated by $2\sqrt[3]{4} – 3$, and dividing $5$ by this number finally gives the factorization

$$5 = (2\sqrt[3]{4} – 3)(6 \sqrt[3]{4} + 8 \sqrt[3]{2} + 9)$$

Here’s some more detail on the trial and error part: we know that $\mathfrak p_1$ is a prime ideal of norm $5$, and we know that $(5) = \mathfrak p_1 \mathfrak p_2$. Since any number with norm $5$ generates a prime ideal, we can conclude by unique factorization of ideals that any such element would have to generate $\mathfrak p_1$. After that, we just look for solutions to the equation $x^3 + 2y^3 + 4z^3 – 6xyz = 5$ (the norm form), and it is easily seen that $x = -3$ and $z = 2$ do the trick.

Note that the factorization you are asking for is impossible: the prime $5$ does not factor as the product of three prime ideals, but as two.

This answer is a try to use the number $1+\sqrt[3]{4}$ as OP did.

As proved by @Starfall, the prime $5$ factors as a product of two primes. With a help from SAGE, the prime factors as
$$5 = (1+\sqrt[3]{4})(1+2\sqrt[3]{2} – \sqrt[3]{4}).$$
One might wonder why the factors look different from @Starfall’s answer. In fact, we have
$$2\sqrt[3]{4} -3 = (1+\sqrt[3]{4})/(1+\sqrt[3]{2}+\sqrt[3]{4})^2,$$
with $1+\sqrt[3]{2}+\sqrt[3]{4}$ being a fundamental unit (i.e. a generator of the free part of unit group) in $\mathbb{Z}[\sqrt[3]{2}]$.
As $\mathbb{Z}[\sqrt[3]{2}]$ is a PID, it is also a UFD. This means each prime ideal is generated by a prime element in the ring. So, the factors $1+\sqrt[3]{4}$ and $1+2\sqrt[3]{2} – \sqrt[3]{4}$ are prime elements in $\mathbb{Z}[\sqrt[3]{2}]$.