Unramified Skolem problems and unramified arithmetic Bertini theorems in positive characteristic

  • Akio Tamagawa


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 §1\S1, 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 §2\S2, we prove the existence of unramified extensions with prescribed local extensions. In §3\S3, 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 §4\S4, 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.