# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Solid $k$-varieties and Henselian fieldsHTML articles powered by AMS MathViewer

by Gustave Efroymson
Trans. Amer. Math. Soc. 170 (1972), 187-195 Request permission

## Abstract:

Let $k$ be a field with a nontrivial absolute value. Define property $( \ast )$ for $k$: Given any polynomial $f(x)$ in $k[x]$ with a simple root $\alpha$ in $k$; then if $g(x)$ is a polynomial near enough to $f(x),g(x)$ has a simple root $\beta$ near $\alpha$. A characterization of fields with property $( \ast )$ is given. If $Y$ is an affine $k$-variety, $Y \subset {\bar k^{(n)}}$, define ${Y_k} = Y \cap {k^{(n)}}$. Define $Y$ to be solid if $I(Y) = I({Y_k})$ in $k[{x_1}, \cdots ,{x_n}]$. If $\pi :Y \to {\bar k^d}$ is a projection induced by Noether normalization, and if $k$ has property $( \ast )$, then $Y$ is a solid $k$-variety if and only if $\pi ({Y_k})$ contains a sphere in ${k^d}$. Using this characterization of solid $k$-varieties and Bertini’s theorem, a dimension theorem is proven.
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