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 Title: Canonical Invariants for Corresponding Residue Systems in P-Adic Fields Author(s): Benson, Steven Rex Doctoral Committee Chair(s): McCulloh, Leon R., Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(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$\sp\prime$)). Issue Date: 1988 Type: Text Description: 50 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988. URI: http://hdl.handle.net/2142/71269 Other Identifier(s): (UMI)AAI8908621 Date Available in IDEALS: 2014-12-16 Date Deposited: 1988
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