<s>[INST] I want know if my logic in showing that $\Bbb Q(\sqrt 2,\sqrt 3) = \Bbb Q (\sqrt 2 + \sqrt 3)$ is correct I want to know if my logic in showing that $\Bbb Q(\sqrt 2,\sqrt 3) = \Bbb Q (\sqrt 2 + \sqrt 3)$ is correct
Now firstly, it seems that $\Bbb Q(\sqrt 2,\sqrt 3)=(\Bbb  Q (\sqrt 2))(\sqrt3){\overset{{\huge\color\red?}}{=}}\{a+b\sqrt2+c\sqrt 3:a,b,c\in \Bbb Q\}$
But from my last question it seems that we need to check this for multiplicative closure, $(a+b\sqrt2 + c\sqrt3)(d+e\sqrt 2 + f\sqrt 3)=ad+ae\sqrt 2+ af\sqrt 3 + 2eb+bf\sqrt 3+cd\sqrt 3+ ce\sqrt2\sqrt3+ 3cf$

So we actually find I believe that:
$\Bbb Q(\sqrt2,\sqrt3)=\{a+b\sqrt 2+c\sqrt 3+ d\sqrt2\sqrt3:a,b,c,d\in \Bbb Q\}$?


I believe that $\Bbb Q(\sqrt 2 +\sqrt 3)\overset{\huge{\color\red ?}}=\{a+b(\sqrt 2+ \sqrt 3):a,b\in \Bbb Q\}$
So let's verify multiplicative closure $(a+b\sqrt 2+ b\sqrt 3)(c+d\sqrt 2+ d\sqrt 3)$
$$=ac+ad\sqrt 2+ad\sqrt 3+bc\sqrt2+2bd+bd\sqrt3+bc\sqrt3+bd\sqrt2\sqrt3+3bd$$
$$=(ac+3bd+2bd)+(ad+bc)\sqrt 2 + (ad+bd+bc)\sqrt 3 +(bd)\sqrt2\sqrt3$$

Since $\sqrt2$ and $\sqrt 3$ and $\sqrt2\sqrt3$ don't share common coefficients, they are linearly independent and hence $\Bbb Q(\sqrt2+\sqrt 3)=\Bbb Q(\sqrt2,\sqrt3)$(from above, i.e we have deduced that:) $\Bbb Q(\sqrt{2}+\sqrt{3})=\{a+b\sqrt2+c\sqrt3:a,b,c\in\Bbb Q\}$.


Is that the correct way to show this? Also this means we have:
$$\begin{matrix}&&\left(\Bbb Q(\sqrt{2}+\sqrt{3})=\Bbb Q(\sqrt2,\sqrt3)\right)\\\\&{\huge\diagup}&&{\huge\diagdown}\\\Bbb Q(\sqrt2)&&&&\Bbb Q(\sqrt 3)\end{matrix}$$
 [/INST] Your belief that
$$\def\Q{\mathbb{Q}}
\Q(\sqrt{2}+\sqrt{3})=\{a+b(\sqrt{2}+ \sqrt{3}):a,b\in \Q\}
$$
is wrong, as well as
$$
\Q(\sqrt{2}+\sqrt{3})=\{a+b\sqrt{2}+ c\sqrt{3}:a,b,c\in \Q\}
$$
The second claim can be immediately dismissed, because this would mean that $[\Q(\sqrt{2}+\sqrt{3}):\Q]=3$, but clearly $\Q(\sqrt{2}+\sqrt{3})\subseteq\Q(\sqrt{2},\sqrt{3})]$ and this extension has degree a divisor of $4$, because
$$
[\Q(\sqrt{2},\sqrt{3}):\Q]=
[\Q(\sqrt{2},\sqrt{3}):\Q(\sqrt{2})]\,
[\Q(\sqrt{2}):\Q]
$$
and both extensions in the right-hand side are of degree either $1$ or $2$ (both $2$, actually, but it's not important).
The first belief is wrong, too: this would mean that $\sqrt{2}+\sqrt{3}$ has a degree two minimal polynomial and this contradicts the fact you have to prove. Indeed
$$
[\Q(\sqrt{2},\sqrt{3}):\Q(\sqrt{2})]=2
$$
because no element of $\Q(\sqrt{2})$ is the square root of $3$:
$$
(a+b\sqrt{2})^2=3
$$
means
$$
a^2+2b^2+2ab\sqrt{2}=3
$$
hence either $a=0$ or $b=0$, leaving either $2b^2=3$ or $a^2=3$; neither equation has a solution in $\Q$.
Now this is a proof that $\Q(\sqrt{2}+\sqrt{3})=\Q(\sqrt{2},\sqrt{3})$. Indeed, we know that $[\Q(\sqrt{2},\sqrt{3}):\Q]=4$ and that
$$
[\Q(\sqrt{2}+\sqrt{3}):\Q]>2
$$
(well, assuming we know that $\sqrt{2}+\sqrt{3}$ is not rational, which is almost obvious). Therefore $[\Q(\sqrt{2}+\sqrt{3}):\Q]$ is a divisor of $4$, but greater than $2$, which leaves just one possibility.

Why doesn't $\sqrt{2}+\sqrt{3}$ have degree $\le2$ over $\Q$? Since $(\sqrt{2}+\sqrt{3})^2=5+2\sqrt{6}$, we see that
$$
\Q\subset\Q(\sqrt{6})\subseteq\Q(\sqrt{2}+\sqrt{3})
$$
and, since $\sqrt{6}$ is clearly irrational (a standard proof based on Eisenstein's criterion applies), we have that the degree of $\sqrt{2}+\sqrt{3}$ (which we already know is a divisor of $4$) can be either $2$ or $4$. If it is $2$, we find that $\sqrt{2}+\sqrt{3}=a+b\sqrt{6}$, for some $a,b\in\Q$. Squaring gives
$$
5+2\sqrt{6}=a^2+6b^2+2ab\sqrt{6}
$$
that implies $ab=1$, so
$$
a^2+\frac{6}{a^2}-5=0
$$
that is,
$$
a^4-5a^2+6=0
$$
or
$$
(a^2-2)(a^2-3)=0
$$
a contradiction.

Of course, the above is much more complicated than the direct observation that
$$
(\sqrt{2}+\sqrt{3})+(\sqrt{2}+\sqrt{3})^{-1}=
\sqrt{2}+\sqrt{3}+\frac{1}{\sqrt{3}+\sqrt{2}}=
\sqrt{2}+\sqrt{3}+\sqrt{3}-\sqrt{2}=2\sqrt{3}
$$
so $\sqrt{3}\in\Q(\sqrt{2}+\sqrt{3})$ and therefore also $\sqrt{2}\in\Q(\sqrt{2}+\sqrt{3})$, ending the proof. However, the main object was to show the issues in your argument.
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