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](http://www.math.harvard.edu/~elkies/Misc/hilbert.pdf) 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?**



Parametrization of unit varieties
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.


