Let be a non-archimedean valuation on a field . Let . Since is ultrametric, is closed under addition and in fact an additive group. The other valuation axioms ensure that is a ring. We call the valuation ring of with respect to the valuation . Note that the field of fractions of is .
Hensel property: Let . Suppose has a simple root . Then has a root and .
Any valued field satisfying the Hensel property is called henselian. The completion of a non-archimedean valued field with respect to the valuation (cf. constructing the reals from the rationals as the completion with respect to the standard metric) is a henselian field.
Every non-archimedean valued field has a unique (up to isomorphism) smallest henselian field containing it. We call the henselisation of .
|Date of creation||2013-03-22 13:28:37|
|Last modified on||2013-03-22 13:28:37|
|Last modified by||mps (409)|
|Defines||residue class field|