I was wondering what elements of the field $Q[\sqrt 2, \sqrt 3]$ look like?
I think that $Q[\sqrt2 + \sqrt3]$ are elements of the form a + b ($\sqrt2 + \sqrt3$), where a and b are in Q.
Are elements of the field $Q[\sqrt 2, \sqrt 3]$ of the form a+b$\sqrt 2$ + c$\sqrt 3$?
Thanks for your help.
Hint: Try multiplying two arbitrary elements of the form $a+b\sqrt{2}+c\sqrt{3}$ with each other. Do you notice a new term appear you didn't have before?
You may do similarly with $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ to see that the proposed basis $\{1,\sqrt{2}+\sqrt{3}\}$ does not work. Think about the degree of the minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb{Q}$, and what relationship this has with the dimension of $\mathbb{Q}(\sqrt{2}+\sqrt{3})$ as a $\mathbb{Q}$-vector space.