Inseparable splitting theory
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- by Richard Rasala
- Trans. Amer. Math. Soc. 162 (1971), 411-448
- DOI: https://doi.org/10.1090/S0002-9947-1971-0284421-2
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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.References
- L. Bégueri, Schéma d’automorphismes. Application a l’étude d’extensions finies radicielles, Bull. Sci. Math. (2) 93 (1965), 89-111.
M. Demazure and A. Grothendieck, Schémas en groupes, fase. 2b, Séminaire Géométrie Algébrique, Inst. Hautes Études Sci., Paris, 1965. MR 34 #7519.
- Günter Pickert, Eine Normalform für endliche reininseparable Körpererweiterungen, Math. Z. 53 (1950), 133–135 (German). MR 37837, DOI 10.1007/BF01162408 Séminaire Heidelberg-Strasbourg, Groupes Algébriques, 1965/66.
- Moss Eisenberg Sweedler, Structure of inseparable extensions, Ann. of Math. (2) 87 (1968), 401–410. MR 223343, DOI 10.2307/1970711
- Oscar Zariski, The concept of a simple point of an abstract algebraic variety, Trans. Amer. Math. Soc. 62 (1947), 1–52. MR 21694, DOI 10.1090/S0002-9947-1947-0021694-1
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- 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