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.

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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

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