TITLE: Parametrization of unit varieties
QUESTION [6 upvotes]: Let $K$ be a number field with integral basis $\{\omega_1,\ldots,\omega_n\}$. Then
$$ \Phi(X_1, \ldots, X_n) = N_{K/{\mathbb Q}}(\omega_1 X_1 + \ldots + \omega_n X_n) $$
is a homogeneous polynomial of degree $n$ with integral coefficients, and the integral points on the affine variety
$$ \Phi(X_1,\ldots,X_n) = 1 $$
correspond to units with norm $+1$ in the ring of integers of $K$.
For quadratic extensions, this "unit variety" is defined by $X_1^2 - mX_2^2 = 1$
(a Pell conic) whenever $m \equiv 2, 3 \bmod 4$ is squarefree; for other extensions of small 
degree it is similarly easy to write down explicit equations.
It is well known that the rational points on the Pell conic can be parametrized. The same thing holds for general cyclic extensions: the 
proof via Hilbert 90 that Pell conics can be parametrized generalizes easily. This suggests the following question:
 Can unit varieties of number fields be parametrized by rational functions with rational coefficients?

REPLY [8 votes]: Thanks to Dror for pointing out a nice geometrical way to think of such varieties.
This paper shows that such varieties are not rational in general:
http://www.math.jussieu.fr/~florence/norm_one.pdf
However, any algebraic torus over a number field $K$ of dimension one or two is rational over $K$. Indeed, the case of dimension one is easy, and the case of dimension two is a theorem of Voskresenskiĭ, but algebraic tori of dimension three and above do not have to be rational over the ground field in general.