Purely inseparable, modular extensions of unbounded exponent

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.