# Henselization of valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks

Same as its importance in commutative algebra. Just to be clear about the definition, for a valued field $K$ with valuation ring $R$, the henselization $K^{\rm{h}}$ is defined to be the valued extension Frac($R^{\rm{h}}$) for the henselization $R^{\rm{h}}$ of $R$ in the sense of commutative algebra (and $R^{\rm{h}}$ is equipped with a preferred valuation extending the one on $R$).

This satisfies good properties as if it were a "completion" of $K$ even though it is (separable) algebraic over $K$, and it can be "approximated" using local-etale extensions of $R$; that is really the point. It satisfies Hensel's Lemma and every finite extension $F$ of $K^{\rm{h}}$ admits a unique valuation (necessarily henselian...) extending the one on $K^{\rm{h}}$ (with associated valuation ring that is the integral closure of $R^{\rm{h}}$ in $F$).

• Thanks Ayanta. I am trying to get the idea about the Henselization and its important in valuation theory. Do you know any text book or any other material where I can read about Henselization? Dec 29 '12 at 5:38
• @Rajnish: To answer your reference question in a useful way it would be helpful to know the reason you are specifically interested in this rather specialized aspect of valuation theory (especially beyond the rank-1 setting). Dec 29 '12 at 5:55
• Thanks Ayanta. I was reading the extension of valuation and suddenly appears the henselization as an immediate extension. That makes kind of hard time to get the idea for me. Dec 30 '12 at 0:00
• @Rajnish: If nothing is being done with it (beyond as an example of an immediate extension) then I recommend focusing on the discretely-valued case (where it's the fixed field of a decomposition group at a place on a separable closure) and ignore the topic until you have a real need to work with it. But if something is being done with it (especially beyond the rank-1 case) then what is that? Anyway, the theory of henselization of local rings beyond dvr's requires a lot of hard work to set up; Raynaud's "Anneaux locaux henseliens" Springer LNM 169 is on exactly this topic. Dec 30 '12 at 0:47