Difference between revisions of "Trace"
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| − | The mapping $\mathrm{Tr}_{K/k}$ of a [[field]] $K$ into a field $k$ (where $K$ is a finite extension of $k$) that sends an element $\alpha \in K$ to the trace of the matrix (cf. [[Trace of a square matrix]]) of the $k$-linear mapping $K \rightarrow K$ sending $\beta \in K$ to $\alpha \beta$. $\mathrm{Tr}_{K/k}$ is a [[homomorphism]] of the additive groups. | + | The mapping $\mathrm{Tr}_{K/k}$ of a [[field]] $K$ into a field $k$ (where $K$ is a finite extension of $k$) that sends an element $\alpha \in K$ to the trace of the matrix (cf. [[Trace of a square matrix]]) of the $k$-linear mapping $K \rightarrow K$ sending $\beta \in K$ to $\alpha \beta$. $\mathrm{Tr}_{K/k}$ is a [[homomorphism]] of the additive groups $K^+ \rightarrow k^+$. |
If $K/k$ is a [[separable extension]], then | If $K/k$ is a [[separable extension]], then | ||
$$ | $$ | ||
| − | \mathrm{Tr}_{K/k} | + | \mathrm{Tr}_{K/k}(\alpha) = \sum_i \sigma_i(\alpha) |
$$ | $$ | ||
where the sum is taken over all $k$-isomorphisms $\sigma_i$ of $K$ into an algebraic closure $\bar k$ of $k$. The trace mapping is transitive, that is, if $L/K$ and $K/k$ are finite extensions, then for any $\alpha \in L$, | where the sum is taken over all $k$-isomorphisms $\sigma_i$ of $K$ into an algebraic closure $\bar k$ of $k$. The trace mapping is transitive, that is, if $L/K$ and $K/k$ are finite extensions, then for any $\alpha \in L$, | ||
Latest revision as of 21:24, 22 December 2014
2020 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]
The mapping $\mathrm{Tr}_{K/k}$ of a field $K$ into a field $k$ (where $K$ is a finite extension of $k$) that sends an element $\alpha \in K$ to the trace of the matrix (cf. Trace of a square matrix) of the $k$-linear mapping $K \rightarrow K$ sending $\beta \in K$ to $\alpha \beta$. $\mathrm{Tr}_{K/k}$ is a homomorphism of the additive groups $K^+ \rightarrow k^+$.
If $K/k$ is a separable extension, then $$ \mathrm{Tr}_{K/k}(\alpha) = \sum_i \sigma_i(\alpha) $$ where the sum is taken over all $k$-isomorphisms $\sigma_i$ of $K$ into an algebraic closure $\bar k$ of $k$. The trace mapping is transitive, that is, if $L/K$ and $K/k$ are finite extensions, then for any $\alpha \in L$, $$ \mathrm{Tr}_{L/k}(\alpha) = \mathrm{Tr}_{K/k}(\mathrm{Tr}_{L/K}(\alpha)) \ . $$
Comments
Especially in older mathematical literature, instead of $\mathrm{Tr}_{K/k}$ one finds $\mathrm{Sp}_{K/k}$ (from the German "Spur" ) as notation for this mapping.
References
| [a1] | N. Jacobson, "Lectures in abstract algebra" , 3. Theory of fields and Galois theory , Springer, reprint (1975) |
| [a2] | N. Jacobson, "Basic algebra" , 1 , Freeman (1985) |
| [a3] | S. Lang, "Algebra" , Addison-Wesley (1965) |
Trace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trace&oldid=35824