How is the axiom of choice related to the well-ordering theorem?

In set theory, the axiom of choice and the well-ordering theorem are mathematically equivalent; assuming one allows for the proof of the other.12

The well-ordering theorem states that every set can be assigned an ordering relation such that every nonempty subset has a least element. Ernst Zermelo first proved this theorem in 1904 by employing the axiom of choice, which asserts that one can simultaneously select an element from each member of an infinite collection of nonempty sets, even without a specific rule or algorithm to do so.123

Because the axiom of choice is nonconstructive—meaning it asserts the existence of a set or function without providing a method to construct it—its relationship with the well-ordering theorem has historically been a subject of controversy. Despite this, the two are considered foundational to modern mathematics, often grouped with Zorn’s lemma as equivalent principles that are widely accepted for their utility in developing classical set theory and the theory of infinite cardinal numbers.12