A valuation criterion for normal basis generators in local fields of characteristic $p$
| dc.creator | Elder, G. Griffith | |
| dc.date | 2008-02-12 | |
| dc.date.accessioned | 2026-07-07T09:20:14Z | |
| dc.date.available | 2026-07-07T09:20:14Z | |
| dc.description | Let $K$ be a complete local field of characteristic $p$ with perfect residue field. Let $L/K$ be a finite, fully ramified, Galois $p$-extension. If $π_L\in L$ is a prime element, and $p'(x)$ is the derivative of $π_L$'s minimal polynomial over $K$, then the relative different $\euD_{L/K}$ is generated by $p'(π_L)\in L$. Let $v_L$ be the normalized valuation normalized with $v_L(L)=\mathbb{Z}$. We show that any element $ρ\in L$ with $v_L(ρ)\equiv -v_L(p'(π_L))-1\bmod[L:K]$ generates a normal basis, $K[{Gal}(L/K)]\cdotρ=L$. This criterion is tight: Given any integer $i$ such that $i\not\equiv -v_L(p'(π_L))-1\bmod[L:K]$, there is a $ρ_i\in L$ with $v_L(ρ_i)=i$ such that $K[{Gal}(L/K)]\cdotρ_i\subsetneq L$. | |
| dc.identifier | https://arxiv.org/abs/0802.1619 | |
| dc.identifier | http://arxiv.org/abs/0802.1619 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/154656 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11S15 | |
| dc.title | A valuation criterion for normal basis generators in local fields of characteristic $p$ | |
| dc.type | text |