UFD’s are integrally closed

Theorem: Every UFD is integrally closedMathworldPlanetmath.

Proof: Let R be a UFD, K its field of fractionsMathworldPlanetmath, uK,u integral over R. Then for some c0,,cn-1R,


Write u=ab,a,bR, where a,b have no non-unit common divisor (which we can assume since R is a UFD). Multiply the above equation by bn to get


Let d be an irreduciblePlanetmathPlanetmath divisor of b. Then d is prime since R is a UFD. Now, d|an since it divides all the other terms and thus (since d is prime) d|a. But a,b have no non-unit common divisors, so d is a unit. Thus b is a unit and hence uR.

Title UFD’s are integrally closed
Canonical name UFDsAreIntegrallyClosed
Date of creation 2013-03-22 15:49:25
Last modified on 2013-03-22 15:49:25
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 13G05