Herbrand's Theorem

Let an ideal class be in ${\mathcal A}$ if it contains an Ideal whose $l$th power is Principal. Let $i$ be an Odd Integer $1\leq i\leq l$ and define $j$ by $i+j=1$. Then ${\mathcal A}_1=\left\langle{e}\right\rangle{}$. If $i\geq 3$ and $l\notdiv B_j$, then ${\mathcal A}_i=\left\langle{e}\right\rangle{}$.

References

Ireland, K. and Rosen, M. ``Herbrand's Theorem.'' §15.3 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 241-248, 1990.