# notation in abstract algebra: $\mathbb{Q}(\sqrt{2})$

What does $$\mathbb{Q}(\sqrt{2})$$ mean? Is it $$\{ a+b\sqrt{2} \mid a, b \in \mathbb{Q} \}$$ ?

• My humble opinion: yes. – manooooh May 8 '19 at 23:17
• See here for ring and field adjunctions. – Bill Dubuque May 8 '19 at 23:18
• Yes, but be careful, $\Bbb Q(\sqrt{2})\neq\{a+b\sqrt{2} \mid a,b\in \Bbb Q \}$. – Dietrich Burde May 9 '19 at 14:52
• hmm, so what is that – Jonny May 9 '19 at 16:33

## 1 Answer

Yes. Formally, $$\mathbb{Q}(a)$$ means “the smallest field containing $$a$$ and $$\mathbb{Q}$$". It turns out that the smallest field containing both $$\mathbb{Q}$$ and $$\sqrt{2}$$ is precisely the field you wrote.

• Good answer!! Why not $\sqrt1$ or $\sqrt0$? – manooooh May 8 '19 at 23:28
• @manooooh what about them? – Yuval Gat May 8 '19 at 23:30
• Why they are not "The smallest field"? Or am I misunderstanding something? – manooooh May 8 '19 at 23:32
• @manooooh Do you know what a field is? – Mark May 8 '19 at 23:32
• Good answer by the way. It is important to note that the definition is what you wrote in the answer. For example $\mathbb{Q}(\pi)$ is not $\{a+b\pi: a,b\in\mathbb{Q}\}$. It's the specific element $\sqrt{2}$ for which $\mathbb{Q}(\sqrt{2})$ can be described in such a simple way. – Mark May 8 '19 at 23:36