# Nilpotent

In mathematics, an element *x* of a ring *R* is called **nilpotent** if there exists some positive integer *n*, called the **index** (or sometimes the **degree**), such that *x*^{n} = 0.

The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras.^{[1]}

An *n*-by-*n* matrix *A* with entries from a field is nilpotent if and only if its characteristic polynomial is *t*^{n}.

More generally, the sum of a unit element and a nilpotent element is a unit when they commute.

A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical: nilpotent elements of ring *R* are precisely those that annihilate all integral domains internal to the ring *R* (that is, of the form *R*/*I* for prime ideals *I*). This follows from the fact that nilradical is the intersection of all prime ideals.

An operand *Q* that satisfies *Q*^{2} = 0 is nilpotent. Grassmann numbers which allow a path integral representation for Fermionic fields are nilpotents since their squares vanish. The BRST charge is an important example in physics.

As linear operators form an associative algebra and thus a ring, this is a special case of the initial definition.^{[4]}^{[5]} More generally, in view of the above definitions, an operator *Q* is nilpotent if there is *n* ∈ **N** such that *Q*^{n} = 0 (the zero function). Thus, a linear map is nilpotent iff it has a nilpotent matrix in some basis. Another example for this is the exterior derivative (again with *n* = 2). Both are linked, also through supersymmetry and Morse theory,^{[6]} as shown by Edward Witten in a celebrated article.^{[7]}

The electromagnetic field of a plane wave without sources is nilpotent when it is expressed in terms of the algebra of physical space.^{[8]} More generally, the technique of microadditivity used to derive theorems makes use of nilpotent or nilsquare infinitesimals, and is part smooth infinitesimal analysis.