The exponent of the smallest power of an ideal which is principal divides the class number

Let \mathbb{O} be the ring of integers in an algebraic number field K of class number k. Let A be an ideal. Show that if k is minimal such that A^k is principal, then k|h.


This k is precisely the order of [A] in the class group of K. The result then follows by Lagrange’s Theorem.

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