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The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673501.png" /> of a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673502.png" /> into a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673503.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673504.png" /> is a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673505.png" /> (cf. [[Extension of a field|Extension of a field]]), that sends an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673506.png" /> to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673507.png" /> that is the determinant of the matrix of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673508.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n0673509.png" /> that takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735011.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735012.png" /> is called the norm of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735013.png" />.
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{{MSC|12F}}
 +
{{TEX|done}}
  
One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735014.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735015.png" />. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735016.png" />,
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The mapping $\def\N{N_{K/k}}\N$ of a
 +
[[Field|field]] $K$ into a field $k$, where $K$ is a finite extension of $k$ (cf.
 +
[[Extension of a field|Extension of a field]]), that sends an element $\def\a{\alpha}\a\in K$ to the element $\N(\a)$ that is the determinant of the matrix of the $k$-linear mapping $K\to K$ that takes $x\in K$ to $\a x$. The element $\N(\a)$ is called the norm of the element $\a$.
  
<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/n06735017.png" /></td> </tr></table>
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One has $\N(\a) = 0$ if and only if $\a = 0$. For any $\def\b{\beta}\a,\b\in K$,
  
that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735018.png" /> induces a homomorphism of the multiplicative groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735019.png" />, which is also called the norm map. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735020.png" />,
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$$\N(\a\b) = \N(\a)\N(\b),$$
 +
that is, $\N$ induces a homomorphism of the multiplicative groups $K^*\to k^*$, which is also called the norm map. For any $\a\in k$,
  
<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/n06735021.png" /></td> </tr></table>
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$$\N(\a)=\a^n,\ \  \textrm{ where } n=[K:k].$$
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The group $\N(K^*)$ is called the norm subgroup of $k^*$, or the group of norms (from $K$ into $k$). If $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ is the characteristic polynomial of $\a\in K$ relative to $k$, then
  
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735022.png" /> is called the norm subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735023.png" />, or the group of norms (from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735024.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735025.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735026.png" /> is the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735027.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735028.png" />, then
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$$\N(\a) = (-1)^n\a_0.$$
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Suppose that $K/k$ is separable (cf.
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[[Separable extension|Separable extension]]). Then for any $\a\in K$,
  
<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/n06735029.png" /></td> </tr></table>
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$$\N(\a) = \prod_{i=1}^n\sigma_i(\a),$$
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where the $\sigma_i$ are all the isomorphisms of $K$ into the
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[[Algebraic closure|algebraic closure]] $\bar k$ of $k$ fixing the elements of $k$.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735030.png" /> is separable (cf. [[Separable extension|Separable extension]]). Then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735031.png" />,
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The norm map is transitive. If $L/K$ and $K/k$ are finite extensions, 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/n/n067/n067350/n06735032.png" /></td> </tr></table>
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$$N_{L/k}(\a)=\N(N_{L/K}(\a))$$
 
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for any $\a\in L$.
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
 
 
 
<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/n06735039.png" /></td> </tr></table>
 
 
 
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067350/n06735040.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1984)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR></table>
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{|
 +
|-
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|valign="top"|{{Ref|BoSh}}||valign="top"| Z.I. Borevich,  I.R. Shafarevich,  "Number theory", Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966) {{MR|0195803}}  {{ZBL|0145.04902}}
 +
|-
 +
|valign="top"|{{Ref|La}}||valign="top"|  S. Lang,  "Algebra", Addison-Wesley  (1984)  {{MR|0799862}} {{MR|0783636}} {{MR|0760079}}  {{ZBL|0712.00001}}
 +
|-
 +
|}

Latest revision as of 08:59, 15 April 2012

2020 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]

The mapping $\def\N{N_{K/k}}\N$ of a field $K$ into a field $k$, where $K$ is a finite extension of $k$ (cf. Extension of a field), that sends an element $\def\a{\alpha}\a\in K$ to the element $\N(\a)$ that is the determinant of the matrix of the $k$-linear mapping $K\to K$ that takes $x\in K$ to $\a x$. The element $\N(\a)$ is called the norm of the element $\a$.

One has $\N(\a) = 0$ if and only if $\a = 0$. For any $\def\b{\beta}\a,\b\in K$,

$$\N(\a\b) = \N(\a)\N(\b),$$ that is, $\N$ induces a homomorphism of the multiplicative groups $K^*\to k^*$, which is also called the norm map. For any $\a\in k$,

$$\N(\a)=\a^n,\ \ \textrm{ where } n=[K:k].$$ The group $\N(K^*)$ is called the norm subgroup of $k^*$, or the group of norms (from $K$ into $k$). If $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ is the characteristic polynomial of $\a\in K$ relative to $k$, then

$$\N(\a) = (-1)^n\a_0.$$ Suppose that $K/k$ is separable (cf. Separable extension). Then for any $\a\in K$,

$$\N(\a) = \prod_{i=1}^n\sigma_i(\a),$$ where the $\sigma_i$ are all the isomorphisms of $K$ into the algebraic closure $\bar k$ of $k$ fixing the elements of $k$.

The norm map is transitive. If $L/K$ and $K/k$ are finite extensions, then

$$N_{L/k}(\a)=\N(N_{L/K}(\a))$$ for any $\a\in L$.

References

[BoSh] Z.I. Borevich, I.R. Shafarevich, "Number theory", Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966) MR0195803 Zbl 0145.04902
[La] S. Lang, "Algebra", Addison-Wesley (1984) MR0799862 MR0783636 MR0760079 Zbl 0712.00001
How to Cite This Entry:
Norm map. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Norm_map&oldid=24335
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