Why does $(5+\sqrt{3})(5-\sqrt{3})$ not conflict with $\mathbb{Q}(\sqrt{3})$ having unique factorization?

Question

My intuition is to try to show that there is some irreducible element $p \in \mathbb{Q}(5)$ that divides $(5+\sqrt{3})(5-\sqrt{3})$, but I'm having trouble finding it. Is there an easier way to do this?

Answer 1

If one is working in $R=\Bbb Z[\sqrt3]$ then one has $$(5+\sqrt3)(5-\sqrt3)=22=2\times11.\tag{*}$$ Is this a failure of unique factorisation in $R$? No, since $$5+\sqrt3=(1+\sqrt3)(-1+2\sqrt3),$$ $$5-\sqrt3=(-1+\sqrt3)(1+2\sqrt3),$$ $$2=(1+\sqrt3)(-1+\sqrt3),$$ and $$11=(1+2\sqrt3)(-1+2\sqrt3).$$ Therefore (*) just shows two ways of combining factors in $$22=(1+\sqrt3)(-1+\sqrt3)(1+2\sqrt3)(-1+2\sqrt3).$$