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Transactions of the American Mathematical Society

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



Purely inseparable, modular extensions of unbounded exponent

Author: Linda Almgren Kime
Journal: Trans. Amer. Math. Soc. 176 (1973), 335-349
MSC: Primary 12F20
MathSciNet review: 0311630
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Abstract: Let K be a purely inseparable extension of a field k of characteristic $ p \ne 0$. Sweedler has shown in [2, p. 403] that if K over k is of finite exponent, then K is modular over k if and only if K can be written as the tensor product of simple extensions of k. This paper grew out of an attempt to find an analogue to this theorem if K is of unbounded exponent over k. The definition of a simple extension is extended to include extensions of the form $ k[x,{x^{1/p}},{x^{1/{p^2}}}, \cdots ][{x^{1/{p^\infty }}}]$. If K is the tensor product of simple extensions, then K is modular. The converse, however, is not true, as several counterexamples in §4 illustrate. Even if we restrict $ [k:{k^p}] < \infty $, the converse is still shown to be false. Given K over k modular, we construct a field $ \cap _{i = 1}^\infty k{K^{{p^i}}} \otimes M( = Q)$ that always imbeds in K where M is the tensor product of simple extensions in the old sense. In general $ K \ne Q$. For K to be the tensor product of simple extensions, we need $ K = Q$, and $ \cap _{i = 1}^\infty k{K^{{p^i}}} = k( \cap _{i = 1}^\infty {K^{{p^i}}})$. If for some finite N, $ k{K^{{p^N}}} = k{K^{{p^{N + 1}}}}$, then we have (by Theorem 11) that $ K = Q$. This finiteness condition guarantees that M is of finite exponent. Should $ \cap _{i = 1}^\infty k{K^{{p^i}}} = k$, then we would have the condition of Sweedler's original theorem. The counterexamples in §4 will hopefully be useful to others interested in unbounded exponent extensions. Of more general interest are two side theorems on modularity. These state that any purely inseparable field extension has a unique minimal modular closure, and that the intersection of modular extensions is again modular.

References [Enhancements On Off] (What's this?)

  • [1] N. Jacobson, Lectures in abstract algebra. Vol. 3: Theory of fields and Galois theory, Van Nostrand, Princeton, N. J., 1964. MR 30 #3087. MR 0172871 (30:3087)
  • [2] M. Sweedler, Structure of inseparable extensions, Ann. of Math. (2) 87 (1968), 401-410. MR 36 #6391. MR 0223343 (36:6391)

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Keywords: Purely inseparable, modular, modular closure, infinite or unbounded exponent, tensor product of simple extensions
Article copyright: © Copyright 1973 American Mathematical Society

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