Is there good intution of the trace map? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:14:21Z http://mathoverflow.net/feeds/question/21062 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21062/is-there-good-intution-of-the-trace-map Is there good intution of the trace map? Tran Chieu Minh 2010-04-12T02:42:11Z 2013-03-08T18:59:21Z <p>I have never understood the trace map,not even after reading <a href="http://mathoverflow.net/questions/13526/geometric-interpretation-of-trace" rel="nofollow">http://mathoverflow.net/questions/13526/geometric-interpretation-of-trace</a>. The problem with many answers in the above discussion is the geometric intuition does not apply to other field. </p> <p>As I don't want this to be closed, let me make the question more precise. Is there a definition of the trace map which</p> <p>1) is basis independent, (there was a definition given by Sridhar Ramesh in the old post).</p> <p>2) explains in an intuitive way why if $L$ is a finite separable extension of $K$, the map $(x,y) \mapsto Tr(xy)$, where $x,y$ are in $L$, is a non degenerated bilinear form on L?</p> http://mathoverflow.net/questions/21062/is-there-good-intution-of-the-trace-map/21064#21064 Answer by Nicolas Ford for Is there good intution of the trace map? Nicolas Ford 2010-04-12T02:59:50Z 2010-04-12T02:59:50Z <p>I don't know if this is what you're looking for, but there's a basis-free definition of the trace in general, outside of the algebraic number theory context -- a linear transformation $V\to V$ corresponds in a natural way to an element of $V\otimes V^\ast$, and the trace map is the map $V\otimes V^\ast\to k$ induced by the bilinear map $V\times V^*\to k$ which sends $(v,f)$ to $f(v)$.</p> <p>But I think the easiest way to see why the trace pairing is nondegenerate for a separable extension is to use bases. Intuitively, nonseparability corresponds to a linear dependency relation among the rows of the matrix because it leads to "repetitions" among the conjugates of some element of the field on top. There's a great (though kind of short) exposition of this stuff in Milne's notes on algebraic number theory: <a href="http://jmilne.org/math/CourseNotes/ant.html" rel="nofollow">http://jmilne.org/math/CourseNotes/ant.html</a></p>