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Let  be a Galois extension, and let  . The norm
 of  is defined to be the product of all the elements of the orbit of  under the group action of the Galois group
 on  ; taken with multiplicities if  is a finite extension.
In the case where  is a finite extension, the norm of  can be defined to be the determinant of the linear transformation
![$ [x]: K \to K$](http://images.planetmath.org:8080/cache/objects/1846/l2h/img11.png "$ [x]: K \to K$") given by
 := xk$](http://images.planetmath.org:8080/cache/objects/1846/l2h/img12.png "$ [x](k) := xk$") , where  is regarded as a vector space over  . This definition does not require that  be Galois, or even that  be a field--for instance, it remains valid when  is a division ring (although  does have to be a field, in order for determinant to be defined). Of course, for finite Galois extensions  , this definition agrees with the previous one, and moreover the formula
holds.
The norm of  is always an element of  , since any element of
 permutes the orbit of  and thus fixes
 .
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"norm" is owned by djao.
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(view preamble)
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 MSC: | 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions) |
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Pending Errata and Addenda
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