Let be a field and . Compute the nilradical of .
We begin with a lemma. (I think I already proved this, but I can’t find it.)
Lemma: Let be a unique factorization domain and an element of , where the are distinct irreducibles. Then . Proof: Suppose . Then for some . Certainly then each must divide , as otherwise no power of is divisible by . Hence . Conversely, we have , so that .
Now is a unique factorization domain, and the lemma applies to .