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.

Tonio