Unramified Skolem problems and unramified arithmetic Bertini theorems in positive characteristic

Abstract

We prove unramified, positive-characteristic versions of theorems of R. S. Rumely [J. Reine Angew. Math. 368, 127--133 (1986; Zbl 0581.14014)] and L. Moret-Bailly [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 2, 161--179 (1989; Zbl 0704.14014)] that generalized Skolem's classical problems, and unramified, positive-characteristic versions of arithmetic Bertini theorems. We also give several applications of these results. We explain the content of each section briefly. In $\S 1$, we investigate the above-mentioned class of polynomials in positive characteristic, namely, superseparable polynomials. The aim here is to control how a superseparable polynomial over a complete discrete valuation field in positive characteristic decomposes. In $\S 2$, we prove the existence of unramified extensions with prescribed local extensions. In $\S 3$, we prove the main results of the present paper, namely, an unramified version of the theorem of Rumely and Moret-Bailly in positive characteristic, and an unramified version of the arithmetic Bertini theorem in positive characteristic. In $\S 4$, we give several remarks and applications of the main results. Some of these applications are essentially new features that only arise after our unramified versions.