I've been wondering if factoring in quadratic rings that are unique factorization domains (principle ideal domains?) is more difficult than factoring integers. Today we can apply the general number field sieve to large integers. Is this applicable to other quadratic rings like Gaussian integers or the other eight imaginary quadratic fields with a class number of one, or the (possibly infinite) number of real quadratic fields?
Are there shortcuts that make it easier to factor in other quadratic fields of class number 1?