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 $\displaystyle \mathbb{F}_{q^n}/\mathbb{F}_q$ is cyclic
of degree n and its Galois group is generated by the Frobenius map $\displaystyle \phi(x):=x^q$ .
Under these conditions Hibert's Theorem 90 says that $\displaystyle \alpha\in\mathbb{F}_{q^n}$ has norm 1 iff
$\displaystyle \exists 0\neq \beta\in\mathbb{F}_{q^n}\,\,s.t.\,\,\alpha=\beta\p hi(\beta)^{-1}$ .
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.
Tonio