where is the symmetric bilinear form associated with , via the polarization identity. Quadratic forms and Clifford algebras in characteristic form an exceptional case. In particular, if it is not true that a quadratic form uniquely determines a symmetric bilinear form satisfying , nor that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not , and are false if this condition is removed. As a quantization of the exterior algebra Clifford algebras are closely related to exterior algebras. Indeed, if then the Clifford algebra is just the exterior algebra . For nonzero there exists a canonical linear isomorphism between and whenever the ground field does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than the exterior product since it makes use of the extra information provided by . The Clifford algebra is a filtered algebra, the associated graded algebra is the exterior algebra. More precisely, Clifford algebras may be thought of as quantizations (cf. quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra. Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras. Universal property and construction Let be a vector space over a field , and let be a quadratic form on . In most cases of interest the field is either the field of real numbers , or the field of complex numbers , or a finite field. A Clifford algebra is a pair , where is a unital associative algebra over and is a linear map satisfying for all in , defined by the following universal property: given any unital associative algebra over and any linear map such that (where denotes the multiplicative identity of ), there is a unique algebra homomorphism such that the following diagram commutes (i.e. such that ): The quadratic form may be replaced by a (not necessarily symmetric) bilinear form that has the property , in which case an equivalent requirement on is When the characteristic of the field is not