Henselian field

Summary

A Henselian field is a valued field in which the conclusion of Hensel's lemma holds, see the definition. In it, a simple root of a polynomial modulo the maximal ideal lifts to a true root. Its valuation ring is a Henselian ring. Complete fields for a discrete valuation, such as the p-adic numbers and fields of formal Laurent series, are Henselian. In a Henselian field the valuation extends in only one way to each algebraic extension, which makes their extension theory clean.