Unique Factorization Pathologies, Revisited

My earlier post on unique factorization pathologies provides a taste of how algebraic number theory is useful as a means of recovering structure in rings without unique factorization. Recall from the post that in a unique factorization domain, an element is both a cube and a square iff it’s a sixth power.

In fact, this is true in any integrally closed integral domain, and in particular in any Dedekind domain. Suppose that r = a^2 = b^3. If a or b is 0, then r is simply 0 = 0^6. If a and b are nonzero, then we can take the element a/b in the fraction field of the ring. This element will satisfy (a/b)^6 = a^6/b^6 = r^3/r^2 = r.

Here is where integral closure comes into play. The element a/b satisfies the polynomial x^6 – r = 0, where r is in the ring. The leading coefficient of the x^6 term is 1, so the polynomial is monic. But the ring is integrally closed, so every element of its fraction field that satisfies a monic polynomial over the ring is itself a member of the ring. Therefore, a/b is in the ring, and r is the sixth power of an element in the ring.

For a counterexample in case the ring is not integrally closed, consider Z[SQRT(8)] = {a + bSQRT(8): a and b are integers}, which would be a Dedekind domain but for integral closure. In that ring, the element 8 is the square of SQRT(8), and the cube of 2, but its obvious sixth root, SQRT(2), is not in the ring, because that would force b to be 1/2.

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