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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Inseparable splitting theory


Author: Richard Rasala
Journal: Trans. Amer. Math. Soc. 162 (1971), 411-448
MSC: Primary 12.45
DOI: https://doi.org/10.1090/S0002-9947-1971-0284421-2
MathSciNet review: 0284421
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Abstract: If L is a purely inseparable field extension of K, we show that, for large enough extensions E of K, the E algebra $ L{ \otimes _K}E$ splits to become a truncated polynomial algebra. In fact, there is a unique smallest extension E of K which splits $ L/K$ and we call this the splitting field $ S(L/K)$ of $ L/K$. Now $ L \subseteq S(L/K)$ and the extension $ S(L/K)$ of K is also purely inseparable. This allows us to repeat the splitting field construction and obtain inductively a tower of fields. We show that the tower stabilizes in a finite number of steps and we study questions such as how soon must the tower stabilize. We also characterize in many ways the case when L is its own splitting field. Finally, we classify all K algebras A which split in a similar way to purely inseparable field extensions.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0284421-2
Keywords: Field, inseparable field extension, splitting, descent theory, truncated polynomial algebra, finite-dimensional algebra, artin ring, derivation, group scheme, automorphism scheme
Article copyright: © Copyright 1971 American Mathematical Society