# surjective function

• March 20th 2011, 02:42 AM
Mike12
surjective function
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 .
• March 20th 2011, 03:07 AM
tonio
Quote:

Originally Posted by Mike12
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 $\mathbb{F}_{q^n}/\mathbb{F}_q$ is cyclic

of degree n and its Galois group is generated by the Frobenius map $\phi(x):=x^q$ .

Under these conditions Hibert's Theorem 90 says that $\alpha\in\mathbb{F}_{q^n}$ has norm 1 iff

$\exists 0\neq \beta\in\mathbb{F}_{q^n}\,\,s.t.\,\,\alpha=\beta\p hi(\beta)^{-1}$ .

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