Introduction to Affine Group Schemes pp 54-61 | Cite as

# Groups of Multiplicative Type

## Abstract

Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an *n* × *n* matrix *g* *separable* if the subalgebra *k*[*g*] of End(*k*^{ n }) is separable. We have of course *k*[*g*] ≃ *k*[*X*]/*p*(*X*) where *p*(*X*) is the minimal polynomial of *g* Separability then holds iff *k*[*g*]⊗\( bar k \) = \( bar k \)[*g*] ⋍ \( bar k \)[*X*]/*p*(*X*) is separable over *k*. This means that *p* has no repeated roots over *k*, which is the familiar criterion for *g* to be diagonalizable over \( bar k \). (We will extend this result in the next section.) Then *p* is separable in the usual Galois theory sense, its roots are in *k*_{ s }, and *g* is diagonalizable over *k*_{ s }.

## Keywords

Abelian Group Hopf Algebra Group Scheme Closed Subgroup Invertible Element## Preview

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