Q is a field, extension fields

Question

What's the difference between Q[√ 3,√ 2] and Q(√ 3,√ 2)?

Q is the rational numbers Please help

Answer 1

They are the same. As for why the different notations, the square brackets denote a polynomial ring, while the round brackets denote a field extension. So $$ Q[\sqrt{3},\sqrt{2}]=\left\{p(\sqrt{3},\sqrt{2}):\, p(x,y)\in \mathbb{Q}[x,y]\right\}=\{a+b\sqrt{3}+c\sqrt{2}+d\sqrt{6}:\,a,b,c,d\in\mathbb{Q}\}, $$ but since $1/\sqrt{3}=\sqrt{3}/3$ (and ditto for $\sqrt{2}$), it does not differ from the field extension.

You would get a difference if you choose transcendental elements (=polynomial ring variables). Here $Q[x]$ is the polynomial ring in $x$, while $Q(x)$ is its fraction field, containing, for example $\frac{x^2+3x+1}{x^4+2x+10}$.