Let E be a splitting field of a separable polynomial over F. Define the Norm N: E-->F by:
N(a) = the product of all q(a) where q is an element of the group Aut(E). I must show that this is a well defined mapping.
The Attempt at a Solution
So I must show that N(a) is an element of a for an arbitrary a in E. To be honest, It suprises me that N(a) would be an element of F, even if a is not an element of F. I mean, by taking the product of all the q(a) where q is an element of the group Aut(E) over F, it seems like it would be entirely possible to get an element that is not in F. Why is this not true? I mean q(a)*q1(a)*...*qr(a) would be in F even if a is not in F? But wouldn't each of qi(a) not be in F? how could the product be?
Hoping somebody can shed a little light on this for me.