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.

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

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