of a field , ring of types of quadratic forms over
The ring of classes of non-degenerate quadratic forms on finite-dimensional vector spaces over with the following equivalence relation: The form is equivalent to the form () if and only if the orthogonal direct sum of the forms and is isometric to the orthogonal direct sum of and for certain neutral quadratic forms and (cf. also Witt decomposition; Quadratic form). The operations of addition and multiplication in are induced by taking the orthogonal direct sum and the tensor product of forms.
Let the characteristic of be different from 2. The definition of equivalence of forms is then equivalent to the following: if and only if the anisotropic forms and which correspond to and (cf. Witt decomposition) are isometric. The equivalence class of the form is said to be its type and is denoted by . The Witt ring, or the ring of types of quadratic forms, is an associative, commutative ring with a unit element. The unit element of is the type of the form . (Here denotes the quadratic form .) The type of the zero form of zero rank, containing also all the neutral forms, serves as the zero. The type is opposite to the type .
The additive group of the ring is said to be the Witt group of the field or the group of types of quadratic forms over . The types of quadratic forms of the form , where is an element of the multiplicative group of , generate the ring . is completely determined by the following relations for the generators:
The Witt ring may be described as the ring isomorphic to the quotient ring of the integer group ring
of the group over the ideal generated by the elements
Here is the residue class of the element with respect to the subgroup .
The Witt ring can often be calculated explicitly. Thus, if is a quadratically (in particular, algebraically) closed field, then ; if is a real closed field, (the isomorphism is realized by sending the type to the signature of the form ); if is a Pythagorean field (i.e. the sum of two squares in is a square) and is not real, then ; if is a finite field, is isomorphic to either the residue ring or , depending on whether or , respectively, where is the number of elements of ; if is a complete local field and its class field has characteristic different from 2, then
An extension of defines a homomorphism of Witt rings for which . If the extension is finite and is of odd degree, is a monomorphism and if, in addition, it is a Galois extension with group , the action of can be extended to and
The general properties of a Witt ring may be described by Pfister's theorem:
1) For any field the torsion subgroup of is -primary;
2) If is a real field and is its Pythagorean closure (i.e. the smallest Pythagorean field containing ), the sequence
is exact (in addition, if , the field is Pythagorean);
3) If is the family of real closures of , the following sequence is exact:
4) If is not a real field, the group is torsion.
A number of other results concern the multiplicative theory of forms. In particular, let be the set of types of quadratic forms on even-dimensional spaces. Then will be a two-sided ideal in , and ; the ideal will contain all zero divisors of ; the set of nilpotent elements of coincides with the set of elements of finite order of and is the Jacobson radical and the primary radical of . The ring is finite if and only if is not real while the group is finite; the ring is Noetherian if and only if the group is finite. If is not a real field, is the unique prime ideal of . If, on the contrary, is a real field, the set of prime ideals of is the disjoint union of the ideal and the families of prime ideals corresponding to orders of :
where runs through the set of prime numbers, and denotes the sign of the element for the order .
If 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 -functor (cf. Algebraic -theory), which play an important role in unitary algebraic -theory.
|||E. Witt, "Theorie der quadratischen Formen in beliebigen Körpern" J. Reine Angew. Math. , 176 (1937) pp. 31–44|
|||N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Addison-Wesley (1973) pp. Chapts. 1–2 (Translated from French)|
|||S. Lang, "Algebra" , Addison-Wesley (1974)|
|||F. Lorenz, "Quadratische Formen über Körpern" , Springer (1970)|
|||O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973)|
|||T.Y. Lam, "The algebraic theory of quadratic forms" , Benjamin (1973)|
|||J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973)|
Given two vector spaces with bilinear forms , , the tensor product is the tensor product with the bilinear form defined by
Witt ring. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Witt_ring&oldid=13336