Field extension : Can one define the notation $\mathbb{Q}(\sqrt{5},\sqrt{2})$?

Question

Can one define the notation Can one define the notation $\mathbb{Q}(\sqrt{5},\sqrt{2})$?

The question is really similar to this one :

field extension-notation problem $\mathbb{Q}(\sqrt{2})=\mathbb{Q}[\sqrt{2}]$

I would say it means "the smallest field that contains all elements of $\mathbb{Q}$ ,contains these two numbers, and is a subfield of $\mathbb{R}$" but i don't understand what in this notation precise that i'm searching a field in $\mathbb{R}$ in particular ,

Answer 1

There is no reason to think about these being subfields of some larger field. $\Bbb Q(\sqrt 5, \sqrt 2)$ means

The smallest field that contains all elements of $\Bbb Q$, and also $\sqrt 2$ and $\sqrt 5$

That's it. No need to specify that it must also be a subfield of something bigger.

We have that $\Bbb R$ contains all the elements of $\Bbb Q$ and also $\sqrt5$ and $\sqrt2$, so naturally it will contain $\Bbb Q(\sqrt5, \sqrt2)$. So we are looking at a subfield of $\Bbb R$. But there is no need to require this to be the case as part of the definition. It just happens automatically. If you did include it in the definition, you'd get into trouble when you try to use that definition to decode $\Bbb Q(i, \sqrt2)$. Or the field $\Bbb Q(x)$ of rational functions in the variable $x$ with rational coefficients.