 Units of Q of square root two

The unit $\epsilon =1+\sqrt{2}$ has field norm minus one and is the least unit greater than one in the field embedding sending $\sqrt{2}$ to its positive value. It is a fundamental unit. The Dirichlet unit theorem therefore gives

Writing ${\epsilon }^n=a_n+b_n\sqrt{2}$, direct multiplication givesFor $n=0,1,2,3,4$, the pairs $(a_n,b_n)$ are $(1,0),(1,1),(3,2),(7,5),(17,12)$, andIn particular, $\eta ={\epsilon }^2=3+2\sqrt{2}$ has field norm one. Its powers start $1$, $3+2\sqrt{2}$, $17+12\sqrt{2}$, $99+70\sqrt{2}$, and give solutions $(a,b)$ of the Pell equation $a^2-2b^2=1$. This turns finding one unit into generating infinitely many exact integer solutions.

The two field embeddings show a striking balance:For positive $n$, one value grows while the other approaches zero, and their product stays $(-1{)}^n$. Taking logarithms of their absolute values converts this multiplication into equally spaced points, the simplest example of the logarithmic embedding of number field units.^[ref]