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?