<s>[INST] Determine the degree of a field extension I have to determine the degree of $\mathbb{Q}\left(\sqrt{2},\sqrt{3}\right)$
over $\mathbb{Q}$ and show that $\sqrt{2}+\sqrt{3}$ is a primitive
element ? Could someone please give me any hints on how to do that
?
 [/INST] Clearly $[\mathbb Q(\sqrt 2):\mathbb Q]\le 2$ becasue of the polynomial $X^2-2$ and $[\mathbb Q(\sqrt 2,\sqrt 3):\mathbb Q(\sqrt 2)]\le 2$ because of the poylnomial $X^2-3$.
In fact, $\sqrt 2\notin \mathbb Q$ implies $[\mathbb Q(\sqrt 2):\mathbb Q]=2$.
We also have $\sqrt 3\notin \mathbb Q(\sqrt 2)$ because $(a+b\sqrt 2)^2 = 3$ implies $(a^2+2b^2) + 2ab\sqrt 2 = 3$, hence $2ab = 0$ and $a^2+2b^2=3$; thus either $a=0$ and $b^2=\frac 32$, or $b=0$ and $a^2=3$. But both $\sqrt{\frac32}$ and $\sqrt 3$ are irrational.
Therefore $[\mathbb Q(\sqrt 2,\sqrt 3):\mathbb Q(\sqrt 2)]=2$ and finally 
$$[\mathbb Q(\sqrt 2,\sqrt 3):\mathbb Q]=4.$$
For the second part , note that $\mathbb Q(\sqrt 2+\sqrt 3)$ contains $(\sqrt 2+\sqrt 3)^2=2+2\sqrt 6+3$, hence also $\sqrt 6$ and $\sqrt6(\sqrt 2+\sqrt 3)=2\sqrt 3+3\sqrt 2$, and finally both $3(\sqrt2+\sqrt 3)-(2\sqrt 3+3\sqrt 2)=\sqrt 2$ and $(2\sqrt 3+3\sqrt 2)-2(\sqrt2+\sqrt 3)=\sqrt 3$.
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