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The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935401.png" /> of a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935402.png" /> into a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935403.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935404.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935405.png" />) that sends an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935406.png" /> to the trace of the matrix (cf. [[Trace of a square matrix|Trace of a square matrix]]) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935407.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935408.png" /> sending <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t0935409.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354010.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354011.png" /> is a [[Homomorphism|homomorphism]] of the additive groups.
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{{TEX|done}}{{MSC|12F}}
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354012.png" /> is a separable extension, then
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354013.png" /></td> </tr></table>
 
 
 
where the sum is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354014.png" />-isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354016.png" /> into an algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354018.png" />. The trace mapping is transitive, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354020.png" /> are finite extensions, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354021.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354022.png" /></td> </tr></table>
 
  
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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.
  
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If $K/k$ is a [[separable extension]], then
 +
$$
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\mathrm{Tr}_{K/k}$ = \sum_i \sigma_i(\alpha)
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$$
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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 $\alpha \in L$,
 +
$$
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\mathrm{Tr}_{L/k}(\alpha) = \mathrm{Tr}_{K/k}(\mathrm{Tr}_{L/K}(\alpha)) \ .
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$$
  
 
====Comments====
 
====Comments====
Especially in older mathematical literature, instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354023.png" /> one finds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093540/t09354024.png" /> (from the German  "Spur" ) as notation for this mapping.
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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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lectures in abstract algebra" , '''3. Theory of fields and Galois theory''' , Springer, reprint  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Jacobson,  "Basic algebra" , '''1''' , Freeman  (1985)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1965)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lectures in abstract algebra" , '''3. Theory of fields and Galois theory''' , Springer, reprint  (1975)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Jacobson,  "Basic algebra" , '''1''' , Freeman  (1985)</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1965)</TD></TR>
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</table>

Revision as of 21:22, 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.

If $K/k$ is a separable extension, then $$ \mathrm{Tr}_{K/k}$ = \sum_i \sigma_i(\alpha) $$ 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)
How to Cite This Entry:
Trace. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trace&oldid=35824
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article