surjectivity of rational points induced by surjective map from affine space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:32:21Z http://mathoverflow.net/feeds/question/100847 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100847/surjectivity-of-rational-points-induced-by-surjective-map-from-affine-space surjectivity of rational points induced by surjective map from affine space ronggang 2012-06-28T07:07:04Z 2012-06-28T08:22:23Z <p>Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic closure of $k$) defined over $k$. Is it true that the restriction of $f$ to $k$ rational points $k^n\to V(k)$ surjective?</p> <p>After it is answered I realized that I simplified what I want to know too much. Please see the comments for the answer for more information.</p> http://mathoverflow.net/questions/100847/surjectivity-of-rational-points-induced-by-surjective-map-from-affine-space/100848#100848 Answer by Olivier Benoist for surjectivity of rational points induced by surjective map from affine space Olivier Benoist 2012-06-28T07:16:50Z 2012-06-28T07:16:50Z <p>No it is not ! Let $f:\mathbb{A}^1\to\mathbb{A}^1$ be defined by $f(x)=x^2$. It is not surjective at the level of $k$-points, because there are elements of $k$ that are not squares.</p>