In order to find the value I took $F =\mathbb{Q}(\sqrt{2})$.
So,$F(\sqrt{6}) = \{a + b\sqrt{6} | a,b \in \mathbb{Q}(\sqrt{2})\}$
So expanding the elements $a$ and $b$ as elements of $\mathbb{Q}(\sqrt{2})$,
$= (a_{1} + b_{1}\sqrt{2}) +(a_{2}+b_{2}\sqrt{2})\sqrt{6}$
Solving a bit more gives me
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In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when corresponding coefficients are equal for each different type of term. The method is used to bring formulas into a desired form.
== Example in real fractions ==
Suppose we want to apply partial fraction decomposition to the expression:
1
x
(
x
...
