University of Illinois Urbana-Champaign

Canonical Invariants for Corresponding Residue Systems in P-Adic Fields

Benson, Steven Rex

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Description

Title
Canonical Invariants for Corresponding Residue Systems in P-Adic Fields
Author(s)
Benson, Steven Rex
Issue Date
1988
Doctoral Committee Chair(s)
McCulloh, Leon R.,
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Abstract
  • Let F be a finite extension of $\doubq\sb{p}$. Let L/F be a normal, totally ramified extension of degree $p\sp{2n}$ with ${\cal B}$ the maximal ideal of ${\cal D}\sb{\rm L}$. Suppose the Hilbert sequence for L/F has a unique breakpoint at $t$. That is, if ${\cal G}$ = Gal(L/F), then its Hilbert sequence is ${\cal G}$ = ${\cal G}\sb0$ = ${\cal G}\sb1$ = $\cdots$ = ${\cal G}\sb{t}\ne{\cal G}\sb{t+1}$ = $\{1\}$. For a subextension K/F of degree $p\sp{n}$ with G = Gal(K/F), we define $\Theta\sbsp{\rm K}{\rm L}$: G $\mapsto$ ${\cal B}\sp{tp\sp{\rm n}}/{\cal B}\sp{tp{\sp\rm n}+1}$ by $\sigma$ $\mapsto$ ${\sigma\pi-\pi}\over{\pi}$ + ${\cal B}\sp{tp\sp{\rm n}+1}$, where $\pi$ is a uniformizer for K/F. If K$\sp\prime$/F is another subextension of degree $p\sp{n}$ with G$\sp\prime$ = Gal(K$\sp\prime$/F), we similarly define $\Theta\sbsp{\rm K\sp\prime}{\rm L}$: G$\sp\prime$ $\to$ ${\cal B}\sp{tp\sp{\rm n}}/{\cal B}\sp{tp\sp{\rm n}+1}.$
  • Define $M\sb{\rm L}$(K,K$\sp\prime$) = max$\{m$: ${\cal D}\sb{\rm K}$ + ${\cal B}\sp{m}$ = ${\cal D}\sb{\rm K\sp\prime}$ + ${\cal B}\sp{m}\}$ and suppose K $\cap$ K$\sp\prime$ = F.
  • If $t$ = 1, we show M$\sb{\rm L}$(K,K$\sp\prime$) = $p\sp{n}$ + $i$ where $i$ is the smallest integer satisfying $\varepsilon\sb{i}(\Theta\sbsp{\rm K}{\rm L}$(G)) $\ne$ $\varepsilon\sb{i}(\Theta\sbsp{\rm K\sp\prime}{\rm L}$(G$\sp\prime$)) in ${\cal B}\sp{itp\sp{\rm n}}/{\cal B}\sp{itp\sp{\rm n}+1}$ and $\varepsilon\sb{i}$ is the $i$th elementary symmetric function. In addition, we show that if $\pi$ and $\pi\sp\prime$ are uniformizers for K/F and K$\sp\prime$/F such that $v\sb{\rm L}$ ($\pi-\pi\sp\prime$) $>$ $v\sb{\rm L}(\pi)$ (= $p\sp{n}$), then $v\sb{\rm L}(\pi-\pi\sp\prime$) = $M\sb{\rm L}$(K,K$\sp\prime$).
  • More generally, if $t$ $<$ $p$, then $M\sb{\rm L}$(K,K$\sp\prime$) $\geq$ ($t$ + 1) $p\sp{n}$-$tp\sp{n-1}$, with equality if and only if $\varepsilon\sb{p\sp{\rm n}-p\sp{\rm n-1}}$($\Theta\sbsp{\rm K}{\rm L}$(G)) $\ne$ $\varepsilon\sb{p\sp{\rm n}-p\sp{\rm n-1}}$($\Theta\sbsp{\rm K\sp\prime}{\rm L}$(G$\sp\prime$)).
Graduation Semester
1988
Type of Resource
text
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