<s>[INST] Find $f$ and $g$ by trial and error and a rigorous proof for showing that $\mathbb{Q}[\sqrt{2} + \sqrt{3}] = \mathbb{Q}[\sqrt{2},\sqrt{3}]$ Here is the question I am trying to solve:
Find polynomials $f(x), g(x) \in \mathbb{Q}[x]$ such that $\sqrt{2} = f(\sqrt{2} + \sqrt{3})$ and $\sqrt{3} = g(\sqrt{2} + \sqrt{3}).$ Deduce the equality of fields: $\mathbb{Q}[\sqrt{2} + \sqrt{3}] = \mathbb{Q}[\sqrt{2},\sqrt{3}].$
And I found the following solution online:

 a formula for $f$ and $g$


 $\mathbb{Q}[\sqrt{2} + \sqrt{3}] = \mathbb{Q}[\sqrt{2},\sqrt{3}]$

My questions are:
1- How did we know that $f$ and $g$ look like that? what are the trials that lead to this?
2- Is there a more rigorous way of proving that  $\mathbb{Q}[\sqrt{2} + \sqrt{3}] \subseteq \mathbb{Q}[\sqrt{2},\sqrt{3}]$? or what is written is enough?
3- the stated reason for showing that  $ \mathbb{Q}[\sqrt{2},\sqrt{3}] \subseteq \mathbb{Q}[\sqrt{2} + \sqrt{3}]$ is not that much clear for me ......could anyone explain it in details for me please?
Note: $F[\alpha]$ is a symbol for subring while $F(\alpha)$ is a symbol for a subfield for any $\alpha$
 [/INST] $(\sqrt 2 + \sqrt 3)^k$ will be a linear combination of $\sqrt 2$, $\sqrt 3$, and $\sqrt 6$ so any polynomial $f(\sqrt 2 + \sqrt 3)$ will yield result of $a\sqrt 6 + b \sqrt 2 +c \sqrt 3 + d$ so we need a polynomial where the yielded values are $a=c=d=0$ and $b = 1$ (and for $g$, $a=b=d=0; c=1$).  And, for simplicity, we want the least power.
Now if we have a power of $k$ and $k+1$ coefficients, $e_k$ we will end up with the $4$ variables, $a,b,c,d$ in $a\sqrt 6 + b \sqrt 2 + c\sqrt 3 + d$ being linear combinations of $e_j$.  That is we will have $4$ equations of combinations of $e_j$ equaling each of $a,b,c,d$.  If we have fewer than $4$ coefficients, $e_j$, we must have linear dependence which are likely to result in inconsistancies.  If we have more than $4$ coefficients we will have superflous solutions.
Best guess is to try a polyinomial of $k+1 = 4$ or of power $k = 3$.
A power of $3$ with $mx^3 + nx^2 + px + q$ will yield $m(2\sqrt 2 + 6\sqrt 3 + 9\sqrt 2+3\sqrt 3) + n(5 + 2\sqrt 6) + p(\sqrt 2 + \sqrt 3) + q$ so we'd need:
$2m +9m+p =11m + p = 1; 6m + 3m+p =9m + p= 0; 2n = 0; 5n+q = 0$ which is a far more promising set of equations.
$n=q=0$ and $11m + p = 1$ and $9m + p =0$. So $m= \frac 12$ and $p=-\frac 92$.
So $f(x) = \frac 12x^3 -\frac 92x$ will yeild $f(\sqrt 2+ \sqrt 3) = \sqrt 2$.
TO find $g$ is much the same but we must solve $11m + p = 0$ and $9m+p = 1$ so $m=-\frac 12$ and $p=\frac {11}2$.
ANd $g(x) = -\frac 12x^3 + \frac {11}2 x$ will yeild $f(\sqrt 2 + \sqrt 3) = \sqrt 3$.
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Note we could have tried powers of $k < 3$ but the $4$ equations and fewer than $4$ unknown all lead to inconsistencies.
$k = 0$ and $f(x) = e_0$ requires $e_0 = \sqrt 2 \in \mathbb Q$.  Impossible. $k = 1$ and $f(x) = e_1x + e_0$ requires $e_1 = 1; e_1=0;e_0=0$.  Impossible.
$k = 2$ and $f(x)=e_2x^2 + e_1x + e_0$ when evaluated for $x=\sqrt 2 + \sqrt 3$ would yield
$e_2(5+2 \sqrt 6) + e_1(\sqrt 2 +\sqrt 3) +e_0$ which to equal $\sqrt 2$ would require $2e_2 = 0; e_1=1; e_1=0; 5e_1+e_0 = 0$ which is of course impossible  ($4$ equations and $3$ unknowns requires linear dependence, and in this case inconsistently so).
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