How to show that $\mathbb Q[\sqrt 2]$ is the smallest subfield of $\mathbb R$ containing $\mathbb Q$ and $\sqrt 2?$

Question

How to show that $\mathbb Q[\sqrt 2]=\{a+b\sqrt 2:a,b\in\mathbb Q\}$ is the smallest subfield of $\mathbb R$ containing $\mathbb Q$ and $\sqrt 2?$

I've shown that $\mathbb Q[\sqrt 2]$ is a subfield of $\mathbb R.$ What about the smallest case?

Answer 1

Is it clear to you what smallest means in this context? This kind of statement means that any other subfield of $\mathbb{R}$ containing $\mathbb{Q}$ and $\sqrt2$ contains $\mathbb{Q}[\sqrt2]$. Can you show this? (It's fairly obvious and certainly the easier containment of the two to show...).