We cover the reduction theory of binary forms. For euclidean number rings R that are discrete in the complex numbers C, we give an algorithm that, given a binary form F over R of degree >= 3,...Show moreWe cover the reduction theory of binary forms. For euclidean number rings R that are discrete in the complex numbers C, we give an algorithm that, given a binary form F over R of degree >= 3, finds the binary with minimal norm or height in the SL2(R) orbit of F. We also generalise this algorithm to discrete number rings that are principal ideal domains (i.e. have class number 1), but give no algorithm to find the smallest one.Show less