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norm (Definition)

Let $ K/F$ be a Galois extension, and let $ x \in K$. The norm $ \operatorname{N}_F^K(x)$ of $ x$ is defined to be the product 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, the norm of $ x$ can be defined to be the determinant of the linear transformation $ [x]: K \to K$ given by $ [x](k) := xk$, where $ K$ is regarded as a vector space over $ F$. This definition does not require that $ K/F$ be Galois, or even that $ K$ be a field--for instance, it remains valid when $ K$ is a division ring (although $ F$ does have to be a field, in order for determinant to be defined). Of course, for finite Galois extensions $ K/F$, this definition agrees with the previous one, and moreover the formula

$\displaystyle \operatorname{N}_F^K(x) := \prod_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) $
holds.

The norm 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{N}_F^K(x)$.



"norm" is owned by djao.
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Cross-references: order, field, division ring, even, vector space, linear transformation, determinant, finite extension, multiplicities, Galois group, group action, orbit, product, Galois extension
There are 70 references to this entry.

This is version 2 of norm, born on 2002-02-07, modified 2003-10-06.
Object id is 1846, canonical name is Norm.
Accessed 11034 times total.

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

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