In particular, \ alpha ^ { p } = a and by the property stated in the paragraph directly above, it follows that F [ \ alpha ] \ supseteq F is a non-trivial purely inseparable extension ( in fact, E = F [ \ alpha ], and so E \ supseteq F is automatically a purely inseparable extension ).
12.
An algebraic extension E \ supset F of fields of non-zero characteristics is a purely inseparable extension if and only if for every \ alpha \ in E \ setminus F, the minimal polynomial of \ alpha over is " not " a separable polynomial, or, equivalently, for every element of, there is a positive integer such that x ^ { p ^ k } \ in F.
13.
Local uniformization in positive characteristic seems to be much harder . proved local uniformization in all characteristic for surfaces and in characteristics at least 7 for 3-folds, and was able to deduce global resolution of singularities in these cases from this . simplified Abhyankar's long proof . extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5 . showed that it is possible to find a local uniformization of any valuation after taking a purely inseparable extension of the function field.
