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We strengthen a result of Michiel Kosters by proving the following theorems: \noindent $(*)$ Let $\phi\colon W\to V$ be a finite surjective morphism of algebraic varieties over an ample field $K$. Suppose $V$ has a simple $K$-rational point ${\bf a}$ such that $\bfa\notin\phi(W(K_\ins))$. Then, ${\rm card}(V(K)\hefresh \phi(W(K))={\rm card}(K)$. \medskip\noindent $(**)$ Let $K$ be an infinite field of positive characteristic and let $f\in K[X]$ be a non-constant monic polynomial. Suppose all zeros of $f$ in $\tilde K$ belong to $K_{\rm ins}\hefresh K$. Then, ${\rm card}(K\hefresh f(K))={\rm card}(K)$.
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참고문헌 (15)
참고문헌 신청
L. Bary-Soroker / 2013 / Geometric and differential Galois theories / Seminaires et Congres :27
J.-L. Colliot-Thelene / 2000 / Rational connectedness and Galois covers of the projective line / Ann. of Math. 151(1):359~373
J. Engler / 2010 / Valued Fields / Springer
M. D. Fried / 2008 / Field Arithmetic, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics / Springer-Verlag :11
U. Gortz / 2010 / Algebraic geometry I / Advanced Lectures in Mathematics
J.-L. Colliot-Thelene / 2000 / Rational connectedness and Galois covers of the projective line / Ann. of Math. 151(1):359~373
J. Engler / 2010 / Valued Fields / Springer
M. D. Fried / 2008 / Field Arithmetic, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics / Springer-Verlag :11
U. Gortz / 2010 / Algebraic geometry I / Advanced Lectures in Mathematics
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