Editing Multiplicative group (section)

==Group scheme of roots of unity==
The '''group scheme of ''n''-th [[roots of unity]]''' is by definition the kernel of the ''n''-power map on the multiplicative group GL(1), considered as a [[group scheme]]. That is, for any integer ''n'' > 1 we can consider the morphism on the multiplicative group that takes ''n''-th powers, and take an appropriate [[fiber product of schemes]], with the morphism ''e'' that serves as the identity.

The resulting group scheme is written μ<sub>''n''</sub> (or <math>\mu\!\!\mu_n</math><ref>{{cite book | last=Milne | first=James S. | title=Étale cohomology | publisher=Princeton University Press | year=1980 | pages=xiii, 66 }}</ref>). It gives rise to a [[reduced scheme]], when we take it over a field ''K'', [[if and only if]] the [[characteristic (field)|characteristic]] of ''K'' does not divide ''n''. This makes it a source of some key examples of non-reduced schemes (schemes with [[nilpotent element]]s in their [[structure sheaf|structure sheaves]]); for example μ<sub>''p''</sub> over a [[finite field]] with ''p'' elements for any [[prime number]] ''p''. 

This phenomenon is not easily expressed in the classical language of algebraic geometry. For example, it turns out to be of major importance in expressing the [[duality theory of abelian varieties]] in characteristic ''p'' (theory of [[Pierre Cartier (mathematician)|Pierre Cartier]]). The [[Galois cohomology]] of this group scheme is a way of expressing [[Kummer theory]].