# Witt ring

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.

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