  In this work, by using Pauli matrices, we introduce four families of
polynomials indexed over the positive integers. These polynomials have rational
or imaginary rational coefficients. It turns out that two of these families are
closely related to classical Lucas and Fibonacci polynomial sequences and hence
to Lucas and Fibonacci numbers. We use one of these families to give a
geometric interpretation of the 200 years old class number problems of Gauss,
which is equivalent to the study of narrow ideal classes in real quadratic
number fields.
