# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Purely inseparable, modular extensions of unbounded exponentHTML articles powered by AMS MathViewer

by Linda Almgren Kime
Trans. Amer. Math. Soc. 176 (1973), 335-349 Request permission

## 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
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 12F20
• Retrieve articles in all journals with MSC: 12F20