UFD’s are integrally closed
Theorem: Every UFD is integrally closed.
Proof: Let be a UFD, its field of fractions, integral over . Then for some ,
Let be an irreducible divisor of . Then is prime since is a UFD. Now, since it divides all the other terms and thus (since is prime) . But have no non-unit common divisors, so is a unit. Thus is a unit and hence .
|Title||UFD’s are integrally closed|
|Date of creation||2013-03-22 15:49:25|
|Last modified on||2013-03-22 15:49:25|
|Last modified by||rm50 (10146)|