On finding elements in inertia groups by reduction modulo \(p\)
Let \(F\) be an algebraic number field, \(P\) a prime ideal of its ring of integers, \({\mathcal O}\) the corresponding localization and \(\pi\) a generator of the maximal ideal of \({\mathcal O}\). Let \(f\in{\mathcal O}[X]\) be monic, without multiple roots and assume that with suitable \(g_ i\in {\mathcal O} [X]\) one has \[ f(X)= g_ 1^{e_ 1} (X) \cdots g_ m^{e_ m} (X)+ \pi h(x), \] with \(e_ i\) not divisible by \(P\). The author describes the cycle structure of a generator of the inertia group of any prime ideal lying over \(P\) in the splitting field of \(f\) in the case when the polynomials \(g_ i\bmod P\) are monic and irreducible in \({\mathcal O}/P [X]\) and none of them divides \(h\bmod P\). For similar results see \textit{B. L. van der Waerden} [Math. Ann. 111, 731-733 (1935; Zbl 0012.24402)], \textit{U. Wegner}, ibid. 738-742 (1935; Zbl 0012.19304)] and \textit{M. Bauer} [Math. Ann. 114, 352-354 (1937; Zbl 0016.24202)].
scientific article; zbMATH DE number 562444