$\mathbf {The \ Problem \ is}:$ Question number $11.$
$\mathbf {My \ approach}:$ Actually, we know $L$ being a finite normal extension, is a splitting field of some polynomial $p(x)$ over $k.$
A hint is given :
Show that $\text{deg} g(x)=\text{deg} h(x).$
But, I am unable to show the above assertion also .
And, how to show that the $k-algebra$ automorphism $\sigma$ of $L$ maps $g(x)$ to $h(x) ?$
A hint is warmly appreciated .Thanks in advance .