View source for Witt ring

''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" />''

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.

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" />.

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:

<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>

<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>

<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>

<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>

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>

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

<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>

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" />.

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

<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>

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

<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>

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;

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

<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>

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);

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:

<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>

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.

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" />:

<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>

<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>

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" />.

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.

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.

====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</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F. Lorenz,   "Quadratische Formen über Körpern" , Springer  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  O.T. O'Meara,   "Introduction to quadratic forms" , Springer  (1973)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  T.Y. Lam,   "The algebraic theory of quadratic forms" , Benjamin  (1973)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  J. Milnor,   D. Husemoller,   "Symmetric bilinear forms" , Springer  (1973)</TD></TR></table>



====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

<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>