# Trace and Norm maps on differential extensions

I'm working through a proof which is rather algebraic, and my abstract algebra is probably only basic to intermediate. I have a differential extension $E/K$ of a differential field $K$, and the proof applies the trace map $Tr$ to both sides of an equation.

I can not find a definition of the trace map in the book, and I think it may be (elementary?) assumed knowledge. What are the definitions of the trace (and norm) maps? I'm guessing it's some kind of morphism? (by which I mean a map with certain properties)

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Every element $a\in E$ affors a linear map $E\to E:x\mapsto ax$; since $E$ is a vector space over $K$ this linear map has a (basis-independent) trace and determinant (the latter called the norm), assuming the extension is finite degree. I don't see that the derivations equipped to the fields are relevant. – anon Aug 9 '12 at 20:18
These tools are basic to algebraic number theory; see here for instance (just culled from a google search). – anon Aug 9 '12 at 20:31
This succinctly facilitated understanding for me: math.uconn.edu/~kconrad/blurbs/galoistheory/tracenorm.pdf – poirot Aug 10 '12 at 7:37

A finite field extension $E/F$ can be considered a vector space over $F$. Then for every $a\in E$, the linear map $x\mapsto ax$ is a vector space endomorphism, and so admits a trace and determinant (the latter here is called the norm), $\mathrm{tr}_{E/F}(a)$ and $N_{E/F}(a)$ respectively (the subscripts may be dropped when the fields are both understood). In fact picking a basis for $E/F$ we can write $x\mapsto ax$ as an explicit matrix with entries in $F$. The trace is $F$-linear and the norm is multiplicative. If $a\in F$ we also have that $\mathrm{tr}(a)=na$ and $N(a)=a^n$, where $n=\dim_{\,F}E$ is $E$'s dimension over $F$.