Almost modular extensions. (Extensions presque modulaires.) (French) Zbl 1140.12004

Summary: Let \(k\) be a commutative field of characteristic \(p>0\). Suppose that \([k:k^p]\) is finite. Let \(K\) be a purely inseparable extension of \(k\). Let \(m/k\) (resp. \(M/K\)) be the smallest extension such that \(K/m\) (resp. \(M/k\)) is modular [see W. C. Waterhouse, Trans. Am. Math. Soc. 211, 39–56 (1975; Zbl 0315.12101)]. We say that \(K/k\) is \(lq\)-modular (resp. \(uq\)-modular) if \(m/k\) (resp. \(K/M)\) is finite. Those two notions are characterized by means of invariants of \(K/k\). We show that any intersection with \(k\) or with \(K\) preserves these two notions. Finally we provide examples of extensions which are non \(lq\)-modular and of extensions which are non \(uq\)-modulaires.


12F15 Inseparable field extensions


Zbl 0315.12101