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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Solid $k$-varieties and Henselian fields
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by Gustave Efroymson PDF
Trans. Amer. Math. Soc. 170 (1972), 187-195 Request permission


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.
  • James Ax, Solving diophantine problems modulo every prime, Ann. of Math. (2) 85 (1967), 161–183. MR 209224, DOI 10.2307/1970438
  • N. Bourbaki, ÉlĂ©ments de mathĂ©matique. Fasc. XXX. AlgĂšbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, ActualitĂ©s Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1308, Hermann, Paris, 1964 (French). MR 0194450
  • D. Dubois and G. Efroymson, A dimension theorem for real primes (submitted). A. Grothendieck with the collaboration of J. DieudonnĂ©, ElĂ©ments de gĂ©omĂ©trie algĂ©brique, Inst. Hautes Études Sci., Bures sur Yvette, Chapter 5 (to appear). A. Grothendieck, Local properties of morphisms, A course given at Harvard University, Cambridge, Mass., 1963.
  • Moshe Jarden, Rational points on algebraic varieties over large number fields, Bull. Amer. Math. Soc. 75 (1969), 603–606. MR 240102, DOI 10.1090/S0002-9904-1969-12257-3
  • Serge Lang, Algebraic numbers, Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto-London, 1964. MR 0160763
  • Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
  • Masayoshi Nagata, On the theory of Henselian rings, Nagoya Math. J. 5 (1953), 45–57. MR 51821
  • Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 170 (1972), 187-195
  • MSC: Primary 14G20; Secondary 13J15
  • DOI:
  • MathSciNet review: 0318159