Difference between revisions of "Norm map"
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<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/n/n067/n067350/n06735032.png" /></td> </tr></table> | <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/n/n067/n067350/n06735032.png" /></td> </tr></table> | ||
− | where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735033.png" /> are all the isomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735034.png" /> into the [[Algebraic closure|algebraic closure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735036.png" />. | + | where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735033.png" /> are all the isomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735034.png" /> into the [[Algebraic closure|algebraic closure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735036.png" /> fixing the elements of $k$. |
The norm map is transitive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735038.png" /> are finite extensions, then | The norm map is transitive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735038.png" /> are finite extensions, then |
Revision as of 21:45, 14 April 2012
The mapping of a field into a field , where is a finite extension of (cf. Extension of a field), that sends an element to the element that is the determinant of the matrix of the -linear mapping that takes to . The element is called the norm of the element .
One has if and only if . For any ,
that is, induces a homomorphism of the multiplicative groups , which is also called the norm map. For any ,
The group is called the norm subgroup of , or the group of norms (from into ). If is the characteristic polynomial of relative to , then
Suppose that is separable (cf. Separable extension). Then for any ,
where the are all the isomorphisms of into the algebraic closure of fixing the elements of $k$.
The norm map is transitive. If and are finite extensions, then
for any .
References
[1] | S. Lang, "Algebra" , Addison-Wesley (1984) |
[2] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) |
Norm map. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Norm_map&oldid=17056