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''of a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980801.png" />, ring of types of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980802.png" />''
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The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980803.png" /> of classes of non-degenerate quadratic forms on finite-dimensional vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980804.png" /> with the following equivalence relation: The form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980805.png" /> is equivalent to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980806.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980807.png" />) if and only if the orthogonal direct sum of the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w0980809.png" /> is isometric to the orthogonal direct sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808011.png" /> for certain neutral quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808013.png" /> (cf. also [[Witt decomposition|Witt decomposition]]; [[Quadratic form|Quadratic form]]). The operations of addition and multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808014.png" /> are induced by taking the orthogonal direct sum and the tensor product of forms.
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Let the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808015.png" /> be different from 2. The definition of equivalence of forms is then equivalent to the following: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808016.png" /> if and only if the anisotropic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808018.png" /> which correspond to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808020.png" /> (cf. [[Witt decomposition|Witt decomposition]]) are isometric. The equivalence class of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808021.png" /> is said to be its type and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808022.png" />. The Witt ring, or the ring of types of quadratic forms, is an associative, commutative ring with a unit element. The unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808023.png" /> is the type of the form . (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808024.png" /> denotes the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808025.png" />.) The type of the zero form of zero rank, containing also all the neutral forms, serves as the zero. The type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808026.png" /> is opposite to the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808027.png" />.
+
''of a field  $  k $,  
 +
ring of types of quadratic forms over  $  k $''
  
The additive group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808028.png" /> is said to be the Witt group of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808029.png" /> or the group of types of quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808030.png" />. The types of quadratic forms of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808032.png" /> is an element of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808034.png" />, generate the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808035.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808036.png" /> is completely determined by the following relations for the generators:
+
The ring  $  W( k) $
 +
of classes of non-degenerate quadratic forms on finite-dimensional vector spaces over  $  k $
 +
with the following equivalence relation: The form  $  f _ {1} $
 +
is equivalent to the form  $  f _ {2} $(
 +
$  f _ {1} \sim f _ {2} $)
 +
if and only if the orthogonal direct sum of the forms  $  f _ {1} $
 +
and  $  g _ {1} $
 +
is isometric to the orthogonal direct sum of $  f _ {2} $
 +
and  $  g _ {2} $
 +
for certain neutral quadratic forms $  g _ {1} $
 +
and  $  g _ {2} $(
 +
cf. also [[Witt decomposition|Witt decomposition]]; [[Quadratic form|Quadratic form]]). The operations of addition and multiplication in  $  W( k) $
 +
are induced by taking the orthogonal direct sum and the tensor product of forms.
  
<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/w/w098/w098080/w09808037.png" /></td> </tr></table>
+
Let the characteristic of  $  k $
 +
be different from 2. The definition of equivalence of forms is then equivalent to the following:  $  f _ {1} \sim f _ {2} $
 +
if and only if the anisotropic forms  $  f _ {1} ^ { a } $
 +
and  $  f _ {2} ^ { a } $
 +
which correspond to  $  f _ {1} $
 +
and  $  f _ {2} $(
 +
cf. [[Witt decomposition|Witt decomposition]]) are isometric. The equivalence class of the form  $  f $
 +
is said to be its type and is denoted by  $  [ f  ] $.
 +
The Witt ring, or the ring of types of quadratic forms, is an associative, commutative ring with a unit element. The unit element of  $  W( k) $
 +
is the type of the form . (Here  $  ( a _ {1} \dots a _ {n} ) $
 +
denotes the quadratic form  $  f( x _ {1} \dots x _ {n} ) = \sum a _ {i} x _ {i}  ^ {2} $.)
 +
The type of the zero form of zero rank, containing also all the neutral forms, serves as the zero. The type  $  [- f  ] $
 +
is opposite to the type  $  [ f  ] $.
  
<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/w/w098/w098080/w09808038.png" /></td> </tr></table>
+
The additive group of the ring  $  W( k) $
 +
is said to be the Witt group of the field  $  k $
 +
or the group of types of quadratic forms over  $  k $.  
 +
The types of quadratic forms of the form  $  ( a) $,
 +
where  $  a $
 +
is an element of the multiplicative group  $  k  ^  \times  $
 +
of  $  k $,
 +
generate the ring  $  W( k) $.  
 +
$  W ( k) $
 +
is completely determined by the following relations for the generators:
  
<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/w/w098/w098080/w09808039.png" /></td> </tr></table>
+
$$
 +
( a)  ( b)  = ( ab),
 +
$$
  
<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/w/w098/w098080/w09808040.png" /></td> </tr></table>
+
$$
 +
( a) + ( b)  = ( a + b) + (( a + b) ab),
 +
$$
 +
 
 +
$$
 +
( a)  ^ {2}  = 1,
 +
$$
 +
 
 +
$$
 +
( a) + (- a)  = 0.
 +
$$
  
 
The Witt ring may be described as the ring isomorphic to the quotient ring of the integer group ring
 
The Witt ring may be described as the ring isomorphic to the quotient ring of the integer group ring
  
<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/w/w098/w098080/w09808041.png" /></td> </tr></table>
+
$$
 +
\mathbf Z [ k  ^  \times  / ( k  ^  \times  )  ^ {2} ]
 +
$$
  
of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808042.png" /> over the ideal generated by the elements
+
of the group $  k  ^  \times  / ( k  ^  \times  )  ^ {2} $
 +
over the ideal generated by the elements
  
<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/w/w098/w098080/w09808043.png" /></td> </tr></table>
+
$$
 +
\overline{1}\; + (- \overline{1}\; ) \  \textrm{ and } \ \
 +
\overline{1}\; + \overline{a}\; - \overline{ {1- a }}\; - \overline{ {( 1 + a) a }}\; \ \
 +
( a \in k  ^  \times  ).
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808044.png" /> is the residue class of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808045.png" /> with respect to the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808046.png" />.
+
Here $  \overline{x}\; $
 +
is the residue class of the element $  x $
 +
with respect to the subgroup $  ( k  ^  \times  )  ^ {2} $.
  
The Witt ring can often be calculated explicitly. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808047.png" /> is a quadratically (in particular, algebraically) closed field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808048.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808049.png" /> is a real closed field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808050.png" /> (the isomorphism is realized by sending the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808051.png" /> to the signature of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808052.png" />); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808053.png" /> is a [[Pythagorean field]] (i.e. the sum of two squares in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808054.png" /> is a square) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808055.png" /> is not real, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808056.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808057.png" /> is a finite field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808058.png" /> is isomorphic to either the residue ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808059.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808060.png" />, depending on whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808061.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808062.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808063.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808064.png" /> is the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808065.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808066.png" /> is a complete local field and its class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808067.png" /> has characteristic different from 2, then
+
The Witt ring can often be calculated explicitly. Thus, if $  k $
 +
is a quadratically (in particular, algebraically) closed field, then $  W( k) \simeq \mathbf Z / 2 \mathbf Z $;  
 +
if $  k $
 +
is a real closed field, $  W( k) \simeq \mathbf Z $(
 +
the isomorphism is realized by sending the type $  [ f  ] $
 +
to the signature of the form $  f  $);  
 +
if $  k $
 +
is a [[Pythagorean field]] (i.e. the sum of two squares in $  k $
 +
is a square) and $  k $
 +
is not real, then $  W( k) \simeq \mathbf Z / 2 \mathbf Z $;  
 +
if $  k $
 +
is a finite field, $  W( k) $
 +
is isomorphic to either the residue ring $  \mathbf Z / 4 \mathbf Z $
 +
or $  ( \mathbf Z / 2 \mathbf Z ) [ t]/ ( t  ^ {2} - 1 ) $,  
 +
depending on whether $  q \equiv 3 $
 +
or $  1 $
 +
$  \mathop{\rm mod}  4 $,  
 +
respectively, where $  q $
 +
is the number of elements of $  k $;  
 +
if $  k $
 +
is a complete local field and its class field $  \overline{k}\; $
 +
has characteristic different from 2, 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/w/w098/w098080/w09808068.png" /></td> </tr></table>
+
$$
 +
W ( k)  \simeq  W ( \overline{k}\; ) [ t] / ( t  ^ {2} - 1).
 +
$$
  
An extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808069.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808070.png" /> defines a homomorphism of Witt rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808071.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808072.png" />. If the extension is finite and is of odd degree, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808073.png" /> is a monomorphism and if, in addition, it is a [[Galois extension|Galois extension]] with group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808074.png" />, the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808075.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808076.png" /> and
+
An extension $  k  ^  \prime  / k $
 +
of $  k $
 +
defines a homomorphism of Witt rings $  \phi : W( k) \rightarrow W( k  ^  \prime  ) $
 +
for which $  [( a _ {1} \dots a _ {n} )] \mapsto [( a _ {1} \dots a _ {n} )] $.  
 +
If the extension is finite and is of odd degree, $  \phi $
 +
is a monomorphism and if, in addition, it is a [[Galois extension|Galois extension]] with group $  G $,  
 +
the action of $  G $
 +
can be extended to $  W( k) $
 +
and
  
<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/w/w098/w098080/w09808077.png" /></td> </tr></table>
+
$$
 +
\phi ( W ( k))  = W ( k  ^  \prime  )  ^ {G} .
 +
$$
  
 
The general properties of a Witt ring may be described by Pfister's theorem:
 
The general properties of a Witt ring may be described by Pfister's theorem:
  
1) For any field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808078.png" /> the torsion subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808080.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808081.png" />-primary;
+
1) For any field $  k $
 +
the torsion subgroup $  W _ {t} ( k) $
 +
of $  W( k) $
 +
is $  2 $-
 +
primary;
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808082.png" /> is a real field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808083.png" /> is its [[Pythagorean closure]] (i.e. the smallest Pythagorean field containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808084.png" />), the sequence
+
2) If $  k $
 +
is a real field and $  k _ {P} $
 +
is its [[Pythagorean closure]] (i.e. the smallest Pythagorean field containing $  k $),  
 +
the sequence
  
<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/w/w098/w098080/w09808085.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  W _ {t} ( k)  \rightarrow  W ( k)  \rightarrow  W ( k _ {P} )
 +
$$
  
is exact (in addition, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808086.png" />, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808087.png" /> is Pythagorean);
+
is exact (in addition, if $  W _ {t} ( k) = 0 $,  
 +
the field $  k $
 +
is Pythagorean);
  
3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808088.png" /> is the family of real closures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808089.png" />, the following sequence is exact:
+
3) If $  \{ k _  \alpha  \} $
 +
is the family of real closures of $  k $,  
 +
the following sequence is exact:
  
<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/w/w098/w098080/w09808090.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  W _ {t} ( k)  \rightarrow  W ( k)  \rightarrow  \prod W ( k _  \alpha  ) ;
 +
$$
  
 
in particular,
 
in particular,
  
4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808091.png" /> is not a real field, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808092.png" /> is torsion.
+
4) If $  k $
 +
is not a real field, the group $  W( k) $
 +
is torsion.
  
A number of other results concern the multiplicative theory of forms. In particular, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808093.png" /> be the set of types of quadratic forms on even-dimensional spaces. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808094.png" /> will be a two-sided ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808096.png" />; the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808097.png" /> will contain all zero divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808098.png" />; the set of nilpotent elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w09808099.png" /> coincides with the set of elements of finite order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080100.png" /> and is the Jacobson radical and the primary radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080101.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080102.png" /> is finite if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080103.png" /> is not real while the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080104.png" /> is finite; the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080105.png" /> is Noetherian if and only if the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080106.png" /> is finite. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080107.png" /> is not a real field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080108.png" /> is the unique prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080109.png" />. If, on the contrary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080110.png" /> is a real field, the set of prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080111.png" /> is the disjoint union of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080112.png" /> and the families of prime ideals corresponding to orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080113.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080114.png" />:
+
A number of other results concern the multiplicative theory of forms. In particular, let $  m $
 +
be the set of types of quadratic forms on even-dimensional spaces. Then $  m $
 +
will be a two-sided ideal in $  W( k) $,  
 +
and $  W( k)/m \simeq \mathbf Z / 2 \mathbf Z $;  
 +
the ideal $  m $
 +
will contain all zero divisors of $  W ( k) $;  
 +
the set of nilpotent elements of $  W( k) $
 +
coincides with the set of elements of finite order of $  m $
 +
and is the Jacobson radical and the primary radical of $  W( k) $.  
 +
The ring $  W( k) $
 +
is finite if and only if $  k $
 +
is not real while the group $  k  ^  \times  / ( k  ^  \times  )  ^ {2} $
 +
is finite; the ring $  W( k) $
 +
is Noetherian if and only if the group $  k  ^  \times  / ( k  ^  \times  )  ^ {2} $
 +
is finite. If $  k $
 +
is not a real field, $  m $
 +
is the unique prime ideal of $  W( k) $.  
 +
If, on the contrary, $  k $
 +
is a real field, the set of prime ideals of $  W( k) $
 +
is the disjoint union of the ideal $  m $
 +
and the families of prime ideals corresponding to orders $  p $
 +
of $  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/w/w098/w098080/w098080115.png" /></td> </tr></table>
+
$$
 +
= \{ {[( a _ {1} \dots a _ {n} )]
 +
} : {\sum  \mathop{\rm sgn} _ {p}  a _ {i} = 0 } \}
 +
,
 +
$$
  
<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/w/w098/w098080/w098080116.png" /></td> </tr></table>
+
$$
 +
P _ {l}  = \{ [( a _ {1} \dots a _ {n} )] : \sum
 +
\mathop{\rm sgn} _ {p}  a _ {i} \equiv 0   \mathop{\rm mod}  l \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080117.png" /> runs through the set of prime numbers, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080118.png" /> denotes the sign of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080119.png" /> for the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080120.png" />.
+
where $  l $
 +
runs through the set of prime numbers, and $  { \mathop{\rm sgn} } _ {p}  a _ {i} $
 +
denotes the sign of the element $  a _ {i} $
 +
for the order $  p $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080121.png" /> is a ring with involution, a construction analogous to that of a Witt ring leads to the concept of the group of a Witt ring with involution.
+
If $  k $
 +
is a ring with involution, a construction analogous to that of a Witt ring leads to the concept of the group of a Witt ring with involution.
  
From a broader point of view, the Witt ring (group) is one of the first examples of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080122.png" />-functor (cf. [[Algebraic K-theory|Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080123.png" />-theory]]), which play an important role in unitary algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080124.png" />-theory.
+
From a broader point of view, the Witt ring (group) is one of the first examples of a $  K $-
 +
functor (cf. [[Algebraic K-theory|Algebraic $  K $-
 +
theory]]), which play an important role in unitary algebraic $  K $-
 +
theory.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Witt, "Theorie der quadratischen Formen in beliebigen Körpern" ''J. Reine Angew. Math.'' , '''176''' (1937) pp. 31–44 {{MR|}} {{ZBL|0015.05701}} {{ZBL|62.0106.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Algebra" , ''Elements of mathematics'' , '''1''' , Addison-Wesley (1973) pp. Chapts. 1–2 (Translated from French) {{MR|2333539}} {{MR|2327161}} {{MR|2325344}} {{MR|2284892}} {{MR|2109105}} {{MR|1994218}} {{MR|1890629}} {{MR|1728312}} {{MR|1727844}} {{MR|1727221}} {{MR|1080964}} {{MR|0979982}} {{MR|0979760}} {{MR|0979493}} {{MR|0928386}} {{MR|0682756}} {{MR|0524568}} {{MR|0573069}} {{MR|0354207}} {{MR|0360549}} {{ZBL|05948094}} {{ZBL|1105.18001}} {{ZBL|1107.13002}} {{ZBL|1107.13001}} {{ZBL|1139.12001}} {{ZBL|1111.00001}} {{ZBL|1103.13003}} {{ZBL|1103.13002}} {{ZBL|1103.13001}} {{ZBL|1017.12001}} {{ZBL|1101.13300}} {{ZBL|0902.13001}} {{ZBL|0904.00001}} {{ZBL|0719.12001}} {{ZBL|0673.00001}} {{ZBL|0666.13001}} {{ZBL|0623.18008}} {{ZBL|0281.00006}} {{ZBL|0279.13001}} {{ZBL|0238.13002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Lorenz, "Quadratische Formen über Körpern" , Springer (1970) {{MR|0282955}} {{ZBL|0211.35303}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) {{MR|}} {{ZBL|0259.10018}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973) {{MR|0396410}} {{ZBL|0259.10019}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) {{MR|0506372}} {{ZBL|0292.10016}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Witt, "Theorie der quadratischen Formen in beliebigen Körpern" ''J. Reine Angew. Math.'' , '''176''' (1937) pp. 31–44 {{MR|}} {{ZBL|0015.05701}} {{ZBL|62.0106.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Algebra" , ''Elements of mathematics'' , '''1''' , Addison-Wesley (1973) pp. Chapts. 1–2 (Translated from French) {{MR|2333539}} {{MR|2327161}} {{MR|2325344}} {{MR|2284892}} {{MR|2109105}} {{MR|1994218}} {{MR|1890629}} {{MR|1728312}} {{MR|1727844}} {{MR|1727221}} {{MR|1080964}} {{MR|0979982}} {{MR|0979760}} {{MR|0979493}} {{MR|0928386}} {{MR|0682756}} {{MR|0524568}} {{MR|0573069}} {{MR|0354207}} {{MR|0360549}} {{ZBL|05948094}} {{ZBL|1105.18001}} {{ZBL|1107.13002}} {{ZBL|1107.13001}} {{ZBL|1139.12001}} {{ZBL|1111.00001}} {{ZBL|1103.13003}} {{ZBL|1103.13002}} {{ZBL|1103.13001}} {{ZBL|1017.12001}} {{ZBL|1101.13300}} {{ZBL|0902.13001}} {{ZBL|0904.00001}} {{ZBL|0719.12001}} {{ZBL|0673.00001}} {{ZBL|0666.13001}} {{ZBL|0623.18008}} {{ZBL|0281.00006}} {{ZBL|0279.13001}} {{ZBL|0238.13002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> F. Lorenz, "Quadratische Formen über Körpern" , Springer (1970) {{MR|0282955}} {{ZBL|0211.35303}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) {{MR|}} {{ZBL|0259.10018}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973) {{MR|0396410}} {{ZBL|0259.10019}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) {{MR|0506372}} {{ZBL|0292.10016}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Given two vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080125.png" /> with bilinear forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080126.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080127.png" />, the tensor product is the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098080/w098080128.png" /> with the bilinear form defined by
+
Given two vector spaces $  V _ {i} $
 +
with bilinear forms $  B _ {i} $,
 +
$  i = 1, 2 $,  
 +
the tensor product is the tensor product $  V _ {1} \otimes V _ {2} $
 +
with the bilinear form 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/w/w098/w098080/w098080129.png" /></td> </tr></table>
+
$$
 +
B( v _ {1} \otimes v _ {2} , w _ {1} \otimes w _ {2} )  = \
 +
B _ {1} ( v _ {1} , w _ {1} ) B _ {2} ( v _ {2} , w _ {2} ) .
 +
$$

Latest revision as of 08:29, 6 June 2020


of a field $ k $, ring of types of quadratic forms over $ k $

The ring $ W( k) $ of classes of non-degenerate quadratic forms on finite-dimensional vector spaces over $ k $ with the following equivalence relation: The form $ f _ {1} $ is equivalent to the form $ f _ {2} $( $ f _ {1} \sim f _ {2} $) if and only if the orthogonal direct sum of the forms $ f _ {1} $ and $ g _ {1} $ is isometric to the orthogonal direct sum of $ f _ {2} $ and $ g _ {2} $ for certain neutral quadratic forms $ g _ {1} $ and $ g _ {2} $( cf. also Witt decomposition; Quadratic form). The operations of addition and multiplication in $ W( k) $ are induced by taking the orthogonal direct sum and the tensor product of forms.

Let the characteristic of $ k $ be different from 2. The definition of equivalence of forms is then equivalent to the following: $ f _ {1} \sim f _ {2} $ if and only if the anisotropic forms $ f _ {1} ^ { a } $ and $ f _ {2} ^ { a } $ which correspond to $ f _ {1} $ and $ f _ {2} $( cf. Witt decomposition) are isometric. The equivalence class of the form $ f $ is said to be its type and is denoted by $ [ f ] $. The Witt ring, or the ring of types of quadratic forms, is an associative, commutative ring with a unit element. The unit element of $ W( k) $ is the type of the form . (Here $ ( a _ {1} \dots a _ {n} ) $ denotes the quadratic form $ f( x _ {1} \dots x _ {n} ) = \sum a _ {i} x _ {i} ^ {2} $.) The type of the zero form of zero rank, containing also all the neutral forms, serves as the zero. The type $ [- f ] $ is opposite to the type $ [ f ] $.

The additive group of the ring $ W( k) $ is said to be the Witt group of the field $ k $ or the group of types of quadratic forms over $ k $. The types of quadratic forms of the form $ ( a) $, where $ a $ is an element of the multiplicative group $ k ^ \times $ of $ k $, generate the ring $ W( k) $. $ W ( k) $ is completely determined by the following relations for the generators:

$$ ( a) ( b) = ( ab), $$

$$ ( a) + ( b) = ( a + b) + (( a + b) ab), $$

$$ ( a) ^ {2} = 1, $$

$$ ( a) + (- a) = 0. $$

The Witt ring may be described as the ring isomorphic to the quotient ring of the integer group ring

$$ \mathbf Z [ k ^ \times / ( k ^ \times ) ^ {2} ] $$

of the group $ k ^ \times / ( k ^ \times ) ^ {2} $ over the ideal generated by the elements

$$ \overline{1}\; + (- \overline{1}\; ) \ \textrm{ and } \ \ \overline{1}\; + \overline{a}\; - \overline{ {1- a }}\; - \overline{ {( 1 + a) a }}\; \ \ ( a \in k ^ \times ). $$

Here $ \overline{x}\; $ is the residue class of the element $ x $ with respect to the subgroup $ ( k ^ \times ) ^ {2} $.

The Witt ring can often be calculated explicitly. Thus, if $ k $ is a quadratically (in particular, algebraically) closed field, then $ W( k) \simeq \mathbf Z / 2 \mathbf Z $; if $ k $ is a real closed field, $ W( k) \simeq \mathbf Z $( the isomorphism is realized by sending the type $ [ f ] $ to the signature of the form $ f $); if $ k $ is a Pythagorean field (i.e. the sum of two squares in $ k $ is a square) and $ k $ is not real, then $ W( k) \simeq \mathbf Z / 2 \mathbf Z $; if $ k $ is a finite field, $ W( k) $ is isomorphic to either the residue ring $ \mathbf Z / 4 \mathbf Z $ or $ ( \mathbf Z / 2 \mathbf Z ) [ t]/ ( t ^ {2} - 1 ) $, depending on whether $ q \equiv 3 $ or $ 1 $ $ \mathop{\rm mod} 4 $, respectively, where $ q $ is the number of elements of $ k $; if $ k $ is a complete local field and its class field $ \overline{k}\; $ has characteristic different from 2, then

$$ W ( k) \simeq W ( \overline{k}\; ) [ t] / ( t ^ {2} - 1). $$

An extension $ k ^ \prime / k $ of $ k $ defines a homomorphism of Witt rings $ \phi : W( k) \rightarrow W( k ^ \prime ) $ for which $ [( a _ {1} \dots a _ {n} )] \mapsto [( a _ {1} \dots a _ {n} )] $. If the extension is finite and is of odd degree, $ \phi $ is a monomorphism and if, in addition, it is a Galois extension with group $ G $, the action of $ G $ can be extended to $ W( k) $ and

$$ \phi ( W ( k)) = W ( k ^ \prime ) ^ {G} . $$

The general properties of a Witt ring may be described by Pfister's theorem:

1) For any field $ k $ the torsion subgroup $ W _ {t} ( k) $ of $ W( k) $ is $ 2 $- primary;

2) If $ k $ is a real field and $ k _ {P} $ is its Pythagorean closure (i.e. the smallest Pythagorean field containing $ k $), the sequence

$$ 0 \rightarrow W _ {t} ( k) \rightarrow W ( k) \rightarrow W ( k _ {P} ) $$

is exact (in addition, if $ W _ {t} ( k) = 0 $, the field $ k $ is Pythagorean);

3) If $ \{ k _ \alpha \} $ is the family of real closures of $ k $, the following sequence is exact:

$$ 0 \rightarrow W _ {t} ( k) \rightarrow W ( k) \rightarrow \prod W ( k _ \alpha ) ; $$

in particular,

4) If $ k $ is not a real field, the group $ W( k) $ is torsion.

A number of other results concern the multiplicative theory of forms. In particular, let $ m $ be the set of types of quadratic forms on even-dimensional spaces. Then $ m $ will be a two-sided ideal in $ W( k) $, and $ W( k)/m \simeq \mathbf Z / 2 \mathbf Z $; the ideal $ m $ will contain all zero divisors of $ W ( k) $; the set of nilpotent elements of $ W( k) $ coincides with the set of elements of finite order of $ m $ and is the Jacobson radical and the primary radical of $ W( k) $. The ring $ W( k) $ is finite if and only if $ k $ is not real while the group $ k ^ \times / ( k ^ \times ) ^ {2} $ is finite; the ring $ W( k) $ is Noetherian if and only if the group $ k ^ \times / ( k ^ \times ) ^ {2} $ is finite. If $ k $ is not a real field, $ m $ is the unique prime ideal of $ W( k) $. If, on the contrary, $ k $ is a real field, the set of prime ideals of $ W( k) $ is the disjoint union of the ideal $ m $ and the families of prime ideals corresponding to orders $ p $ of $ k $:

$$ P = \{ {[( a _ {1} \dots a _ {n} )] } : {\sum \mathop{\rm sgn} _ {p} a _ {i} = 0 } \} , $$

$$ P _ {l} = \{ [( a _ {1} \dots a _ {n} )] : \sum \mathop{\rm sgn} _ {p} a _ {i} \equiv 0 \mathop{\rm mod} l \} , $$

where $ l $ runs through the set of prime numbers, and $ { \mathop{\rm sgn} } _ {p} a _ {i} $ denotes the sign of the element $ a _ {i} $ for the order $ p $.

If $ k $ is a ring with involution, a construction analogous to that of a Witt ring leads to the concept of the group of a Witt ring with involution.

From a broader point of view, the Witt ring (group) is one of the first examples of a $ K $- functor (cf. Algebraic $ K $- theory), which play an important role in unitary algebraic $ K $- theory.

References

[1] E. Witt, "Theorie der quadratischen Formen in beliebigen Körpern" J. Reine Angew. Math. , 176 (1937) pp. 31–44 Zbl 0015.05701 Zbl 62.0106.02
[2] N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Addison-Wesley (1973) pp. Chapts. 1–2 (Translated from French) MR2333539 MR2327161 MR2325344 MR2284892 MR2109105 MR1994218 MR1890629 MR1728312 MR1727844 MR1727221 MR1080964 MR0979982 MR0979760 MR0979493 MR0928386 MR0682756 MR0524568 MR0573069 MR0354207 MR0360549 Zbl 05948094 Zbl 1105.18001 Zbl 1107.13002 Zbl 1107.13001 Zbl 1139.12001 Zbl 1111.00001 Zbl 1103.13003 Zbl 1103.13002 Zbl 1103.13001 Zbl 1017.12001 Zbl 1101.13300 Zbl 0902.13001 Zbl 0904.00001 Zbl 0719.12001 Zbl 0673.00001 Zbl 0666.13001 Zbl 0623.18008 Zbl 0281.00006 Zbl 0279.13001 Zbl 0238.13002
[3] S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[4] F. Lorenz, "Quadratische Formen über Körpern" , Springer (1970) MR0282955 Zbl 0211.35303
[5] O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) Zbl 0259.10018
[6] T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973) MR0396410 Zbl 0259.10019
[7] J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) MR0506372 Zbl 0292.10016

Comments

Given two vector spaces $ V _ {i} $ with bilinear forms $ B _ {i} $, $ i = 1, 2 $, the tensor product is the tensor product $ V _ {1} \otimes V _ {2} $ with the bilinear form defined by

$$ B( v _ {1} \otimes v _ {2} , w _ {1} \otimes w _ {2} ) = \ B _ {1} ( v _ {1} , w _ {1} ) B _ {2} ( v _ {2} , w _ {2} ) . $$

How to Cite This Entry:
Witt ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Witt_ring&oldid=49233
This article was adapted from an original article by A.V. MikhalevA.I. NemytovV.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article