Suppose $\displaystyle E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) $. Does the map defined by $\displaystyle \sigma(a+b \sqrt{2} + c \sqrt{3}+d \sqrt{6}) = a+ b\sqrt{2}-c \sqrt{3}- d \sqrt{6} $ leave something fixed?

First of all, $\displaystyle E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) $ is the set $\displaystyle \mathbb{Q} $ with the elements $\displaystyle \sqrt{2} $ and $\displaystyle \sqrt{3} $ added in? It is an autmomorphism of E?