# Exterior Algebras

Since V2.15 (December 2008), Magma has a special type for *exterior algebras*. Such an algebra is skew-commutative and is a quotient of the free algebra $K\langle x_1, \ldots, x_n\rangle$ by the relations $x_i^2 = 0$ and $x_i x_j = -x_j x_i$ for $1 \leq i,j \leq n$, $i \not= j$. Because of these relations, elements of the algebra can be written in terms of commutative monomials in the variables (via a collection algorithm), and the associated algorithms are much more efficient than for the general noncommutative case. Also, a Gröbner basis of an ideal of an exterior algebra is always finite (in fact, the whole exterior algebra has dimension $2^n$ as a $K$-vector space). Exterior algebras may be constructed with the [`ExteriorAlgebra`](algebras.md#function-fpa-exterioralgebra) function below, and all operations applicable to general FP algebras are also applicable to them (so will not be duplicated here). Furthermore, modules over exterior algebras are also allowed: see Chapter [Modules over Multivariate Rings](../../CommutativeAlgebra/ModulesOverMultivariateRings/index-modules-over-multivariate-rings.md#pmod-chap) for details.