Algorithms for $p$-rationality and for $p$-saturation of units

Let $K$ be a number field. We give new practical algorithms that determine whether $K$ is $p$-rational. For real cyclotomic fields $K=\mathbb{Q}(ζ_{n})^{+}$, our method avoids computing either the class number or the full unit group. We also use the notion of $p$-rationality to explain the practical