if N is a norm function from Fq^n to Fq . how can I prove that for alpha belongs to Fq^n star , N(alpha) =1 iff there exists an element Beta belongs to Fq^n such that alpha is equal to (Beta)^1-q .
It's hard to understand what you mean (learn to use LaTeX! In the forum' section
"Help with LaTeX"), but I think that what you want is Hilbert's Theorem 90, since is cyclic
of degree n and its Galois group is generated by the Frobenius map .
Under these conditions Hibert's Theorem 90 says that has norm 1 iff
Now, I'm not sure but it seems this is what you want. You can either google the result or look for it
in any decente Galois Theory book.