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(kn) 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 .
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© 1979 Springer-Verlag New York Inc.
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Waterhouse, W.C. (1979). Groups of Multiplicative Type. In: Introduction to Affine Group Schemes. Graduate Texts in Mathematics, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-6217-6_7
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DOI: https://doi.org/10.1007/978-1-4612-6217-6_7
Publisher Name: Springer, New York, NY
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