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trace (Definition)

Let $ K/F$ be a Galois extension, and let $ x \in K$. The trace $ \operatorname{Tr}_F^K(x)$ of $ x$ is defined to be the sum of all the elements of the orbit of $ x$ under the group action of the Galois group $ \operatorname{Gal}(K/F)$ on $ K$; taken with multiplicities if $ K/F$ is a finite extension.

In the case where $ K/F$ is a finite extension,

$\displaystyle \operatorname{Tr}_F^K(x) := \sum_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) $

The trace of $ x$ is always an element of $ F$, since any element of $ \operatorname{Gal}(K/F)$ permutes the orbit of $ x$ and thus fixes $ \operatorname{Tr}_F^K(x)$.

The name “trace” derives from the fact that, when $ K/F$ is finite, the trace of $ x$ is simply the trace of the linear transformation $ T: K \longrightarrow K$ of vector spaces over $ F$ defined by $ T(v) := xv$.



"trace" is owned by djao.
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examples of trace and norm (Example) by polarbear
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Cross-references: vector spaces, linear transformation, finite extension, multiplicities, Galois group, group action, orbit, sum, Galois extension
There are 17 references to this entry.

This is version 4 of trace, born on 2002-02-07, modified 2005-04-03.
Object id is 1845, canonical name is Trace2.
Accessed 4980 times total.

Classification:
AMS MSC12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)

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