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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803501.png" /> be a finite-dimensional [[Central simple algebra|central simple algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803502.png" />. A finite extension field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803503.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803504.png" /> is a splitting field for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803505.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803506.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803507.png" />-algebras for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803508.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r0803509.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035010.png" />-algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035011.png" />-matrices. Choose an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035012.png" />. The reduced norm mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035013.png" /> is now defined by
 
  
<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/r/r080/r080350/r08035014.png" /></td> </tr></table>
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{{MSC|16H05}}
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{{TEX|done}}
  
and the reduced trace mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035015.png" /> is similarly defined by
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Let $A$ be a finite-dimensional
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[[Central simple algebra|central simple algebra]] over $k$. A finite extension field $K$ of $k$ is a splitting field for $A$ if $\def\M{\textrm{M}} A\otimes_k K\simeq \M_m(K)$ as $K$-algebras for some $m$. Here $\M_m(K)$ is the $K$-algebra of $(m\times m)$-matrices. Choose an isomorphism $\def\phi{\varphi} \phi:A\otimes_k K \to \M_m(K)$. The reduced norm mapping $\def\Nrd{\textrm{Nrd}} \Nrd_{A/k}:A\to k$ is now defined by
  
<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/r/r080/r080350/r08035016.png" /></td> </tr></table>
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$$\Nrd_{A/k}(a) = \det(\phi(a\otimes 1)),$$
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and the reduced trace mapping $\def\Trd{\textrm{Trd}} \Trd_{A/k}(a)$ is similarly defined by
  
One checks that the right-hand sides of these equations are indeed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035017.png" /> (and not just in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035018.png" />) and that the definitions are independent of the choices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035020.png" />.
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$$\Trd_{A/k}(a) = \textrm{trace}(\phi(a\otimes1)).$$
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One checks that the right-hand sides of these equations are indeed in $k$ (and not just in $K$) and that the definitions are independent of the choices of $\phi$ and $K$.
  
The reduced norm is multiplicative, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035021.png" /> is invertible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035022.png" />. The reduced trace is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035023.png" /> vector spaces, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035024.png" /> defines a non-degenerate bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035025.png" />.
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The reduced norm is multiplicative, and $a\in A$ is invertible if and only if $\Nrd_{A/k}(a) \ne 0$. The reduced trace is a homomorphism of $k$-vector spaces, and $(x,y)\mapsto \Trd_{A/k}(xy)$ defines a non-degenerate bilinear form on $A$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Bass,  "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035026.png" />-theory" , Benjamin  (1967)  pp. 152ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.J. Hahn,  O.T. O'Meara,  "The classical groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080350/r08035027.png" />-theory" , Springer  (1979)  pp. §2.2D</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Ba}}||valign="top"| H. Bass,  "Algebraic K-theory", Benjamin  (1967)  pp. 152ff {{MR|0279159}}  {{ZBL|0226.13006}}   
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|-
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|valign="top"|{{Ref|HaOM}}||valign="top"| A.J. Hahn,  O.T. O'Meara,  "The classical groups and $K$-theory", Springer  (1979)  pp. §2.2D   {{MR|1007302}}  {{ZBL|0683.20033}} 
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|-
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|}

Latest revision as of 19:38, 5 March 2012

2010 Mathematics Subject Classification: Primary: 16H05 [MSN][ZBL]

Let $A$ be a finite-dimensional central simple algebra over $k$. A finite extension field $K$ of $k$ is a splitting field for $A$ if $\def\M{\textrm{M}} A\otimes_k K\simeq \M_m(K)$ as $K$-algebras for some $m$. Here $\M_m(K)$ is the $K$-algebra of $(m\times m)$-matrices. Choose an isomorphism $\def\phi{\varphi} \phi:A\otimes_k K \to \M_m(K)$. The reduced norm mapping $\def\Nrd{\textrm{Nrd}} \Nrd_{A/k}:A\to k$ is now defined by

$$\Nrd_{A/k}(a) = \det(\phi(a\otimes 1)),$$ and the reduced trace mapping $\def\Trd{\textrm{Trd}} \Trd_{A/k}(a)$ is similarly defined by

$$\Trd_{A/k}(a) = \textrm{trace}(\phi(a\otimes1)).$$ One checks that the right-hand sides of these equations are indeed in $k$ (and not just in $K$) and that the definitions are independent of the choices of $\phi$ and $K$.

The reduced norm is multiplicative, and $a\in A$ is invertible if and only if $\Nrd_{A/k}(a) \ne 0$. The reduced trace is a homomorphism of $k$-vector spaces, and $(x,y)\mapsto \Trd_{A/k}(xy)$ defines a non-degenerate bilinear form on $A$.

References

[Ba] H. Bass, "Algebraic K-theory", Benjamin (1967) pp. 152ff MR0279159 Zbl 0226.13006
[HaOM] A.J. Hahn, O.T. O'Meara, "The classical groups and $K$-theory", Springer (1979) pp. §2.2D MR1007302 Zbl 0683.20033
How to Cite This Entry:
Reduced norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reduced_norm&oldid=13357