Is it obvious that $\mathbb Q(\sqrt 3+\sqrt 5)=\mathbb Q(\sqrt 3,\sqrt 5)$?

Question

Is it obvious that $\mathbb Q(\sqrt 3+\sqrt 5)=\mathbb Q(\sqrt 3,\sqrt 5)$ ? If not how can I show it ?

Answer 1

The first inclusion is obvious because $\mathbb{Q}(\sqrt{3},\sqrt{5})$ is closed under addition. So $\mathbb{Q}(\sqrt{3} +\sqrt{5}) \subseteq \mathbb{Q}(\sqrt{3} ,\sqrt{5})$.

To second inclusion observe that $$(\sqrt{3} +\sqrt{5})(\sqrt{3} -\sqrt{5}) = 3 - 5 = -2 \Rightarrow \sqrt{5} -\sqrt{3} = \frac{-2}{\sqrt{5} +\sqrt{3}}$$

then $\sqrt{5} -\sqrt{3} \in \mathbb{Q}(\sqrt{3} +\sqrt{5})$. Now use sum and subtraction to find that $\sqrt{3}, \sqrt{5} \in \mathbb{Q}(\sqrt{3} +\sqrt{5})$.

Can you take it from here?

Answer 2

It is obvious if the facts below are obvious:

• $\mathbb Q(\sqrt 3,\sqrt 5)$ has degree 4

• $\mathbb Q(\sqrt 3+\sqrt 5)$ has degree 4

• $\mathbb Q(\sqrt 3+\sqrt 5) \subseteq \mathbb Q(\sqrt 3,\sqrt 5)$

Of these, only the last one is really obvious.

Answer 3

$(\sqrt3+\sqrt 5)^2\in \mathbb Q(\sqrt 3+\sqrt 5)$

$\implies 8+2\sqrt3.\sqrt 5\in \mathbb Q(\sqrt 3+\sqrt 5)$

$\implies \sqrt3.\sqrt 5\in \mathbb Q(\sqrt 3+\sqrt 5)$

$\implies (\sqrt3+\sqrt 5).\sqrt3.\sqrt 5\in \mathbb Q(\sqrt 3+\sqrt 5)$

$\implies 3\sqrt 5+5\sqrt3\in \mathbb Q(\sqrt 3+\sqrt 5)$

$3(\sqrt3+\sqrt 5)\in \mathbb Q(\sqrt 3+\sqrt 5)$

$\implies 2\sqrt 3\in \mathbb Q(\sqrt 3+\sqrt 5)$

$\implies \sqrt 3\in \mathbb Q(\sqrt 3+\sqrt 5)$

similarly $\sqrt 5\in \mathbb Q(\sqrt 3+\sqrt 5)$

Thus $\mathbb Q(\sqrt 3+\sqrt 5)\subseteq \mathbb Q(\sqrt3,\sqrt5)$

The other part is obvious