is there any subset of the reals that is not a Unique Factorization Domain?

Question

Is there any subset of the real numbers that is not a Unique Factorization Domain? (i.e. where within that subset, a "prime" is a number that cannot be written as a product of any numbers in that set except itself and 1, and where there is at least one number that can be written as the product of two different sets of primes).

I usually introduce recreational math students to the concept of a non-UFD by showing them the set of all numbers a + ibsqrt(5). So I wondered if you can do it without complex numbers, i.e. find a subset of the reals that is a non-UFD, or prove that it's impossible.

Does this answer your question? When is $\mathbb{Z}[\sqrt{d}]$ not an UFD (for $d\ge 1$)? — Dietrich Burde, Nov 04 '21 at 20:51
Then take $\Bbb Z[\sqrt{5}]$ instead of $\Bbb Z[\sqrt{-5}]$. Both are not UFD, and the first one consists of real numbers only. — Dietrich Burde, Nov 04 '21 at 20:54
Note that it's somehow easier to show that $\Bbb Z[\sqrt{-5}]$ is not UFD, because there are not many units. — WhatsUp, Nov 04 '21 at 20:55
@WhatsUp Well, a factorial ring would integrally closed, and $\Bbb Z[\sqrt{5}]$ is not integrally closed. This is easy. — Dietrich Burde, Nov 04 '21 at 20:56
@DietrichBurde I suppose the audience don't have knowledge of e.g. integrally closed. But yes it's not the maximal order. — WhatsUp, Nov 04 '21 at 20:59
Here is an elementary proof that $\mathbb Z[\sqrt5]$ is not factorial. — Dietrich Burde, Nov 04 '21 at 21:07

score 3 · Accepted Answer · answered Nov 04 '21 at 20:59

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You asked the subset of real numbers, so I'll give an example $\mathbb Z[\sqrt5]$.

In that case, $2, -2, 1\pm\sqrt5$ is all prime, but $-2\times2=(1-\sqrt5)(1+\sqrt5)$.

So, $\mathbb Z[\sqrt5]$ is surely subring of $\mathbb R$, but it is not UFD.

answered Nov 04 '21 at 20:59

MH.Lee

5,568

Davide Trono · Answer 2 · 2021-11-04T21:08:16.917

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Consider $\mathbb{Z}[\sqrt{10}]$, then $9=3\cdot3=(1+\sqrt{10})(\sqrt{10}-1)$ and $3, 1+\sqrt{10}, \sqrt{10} - 1$ are all prime numbers in this ring (although the proof of them being prime is far from obvious). Also, please notice in this context 'subset' doesn't make much sense.

edited Nov 04 '21 at 21:08

answered Nov 04 '21 at 20:56

Davide Trono

1,431
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I see, then let me correct my mistake. – Davide Trono Nov 04 '21 at 20:59
"Far from obvious" depends on the prerequisites one has. Anyway, it is shown in this duplicate here. – Dietrich Burde Nov 04 '21 at 21:10

is there any subset of the reals that is not a Unique Factorization Domain?

2 Answers2