On the main invariant of elements algebraic over a Henselian valued field

Kamal, Aghigh ; Khanduja, Sudesh K. (2002) On the main invariant of elements algebraic over a Henselian valued field Proceedings of the Edinburgh Mathematical Society, 45 (1). pp. 219-227. ISSN 0013-0915

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Related URL: http://dx.doi.org/10.1017/S0013091500000936

Abstract

Let $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.

Item Type:Article
Source:Copyright of this article belongs to Cambridge University Press.
Keywords:Valued Fields; Valuations and their Generalizations; Non-archimedean Valued Fields
ID Code:69883
Deposited On:19 Nov 2011 11:06
Last Modified:19 Nov 2011 11:06

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