# henselian field

Let $|\cdot |$ be a non-archimedean valuation on a field $K$. Let
$V=\{x:|x|\le 1\}$. Since $|\cdot |$ is ultrametric, $V$ is closed under
addition and in fact an additive group^{}. The other valuation^{} axioms
ensure that $V$ is a ring. We call $V$ the *valuation ring ^{}* of
$K$ with respect to the valuation $|\cdot |$. Note that the field of
fractions

^{}of $V$ is $K$.

The set $$ is a maximal ideal^{} of $V$. The factor
$R:=V/\mu $ is called the *residue field ^{}* or the

*residue class field*.

The map $\mathrm{res}:V\to V/\mu $ given by $x\mapsto x+\mu $ is called the
*residue map*. We extend the definition of the residue map to
sequences of elements from $V$, and hence to $V[X]$ so that if $f(X)\in V[X]$ is given by ${\sum}_{i\le n}{a}_{i}{X}^{i}$ then $\mathrm{res}(f)\in R[X]$ is given by ${\sum}_{i\le n}\mathrm{res}(ai){X}^{i}$.

Hensel property: Let $f(x)\in V[x]$. Suppose $\mathrm{res}(f)(x)$ has a simple root $e\in k$. Then $f(x)$ has a root ${e}^{\prime}\in V$ and $\mathrm{res}({e}^{\prime})=e$.

Any valued field satisfying the Hensel property is called
*henselian*. The completion of a non-archimedean valued field $K$
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 $K$ has a unique (up to
isomorphism^{}) smallest henselian field ${K}^{h}$ containing it. We call
${K}^{h}$ the *henselisation* of $K$.

Title | henselian field |

Canonical name | HenselianField |

Date of creation | 2013-03-22 13:28:37 |

Last modified on | 2013-03-22 13:28:37 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 9 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 13F30 |

Classification | msc 13A18 |

Classification | msc 11R99 |

Classification | msc 12J20 |

Related topic | Valuation |

Related topic | ValuationDomainIsLocal |

Related topic | ValuationRingOfAField |

Defines | valuation ring |

Defines | residue field |

Defines | residue class field |

Defines | Hensel property |

Defines | henselian |

Defines | henselisation |