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?**