{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71269"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71269","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Canonical Invariants for Corresponding Residue Systems in P-Adic Fields","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$\\sp\\prime$ $\\to$ ${\\cal B}\\sp{tp\\sp{\\rm n}}/{\\cal B}\\sp{tp\\sp{\\rm n}+1}.$","abstract_html":"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 <span class=\"etd-inline-math\">&sigma;</span> $\\mapsto$ <span class=\"etd-inline-math\">{&sigma;&pi;-&pi;}\\over{&pi;}</span> + ${\\cal B}\\sp{tp\\sp{\\rm n}+1}$, where <span class=\"etd-inline-math\">&pi;</span> 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$\\sp\\prime$ $\\to$ ${\\cal B}\\sp{tp\\sp{\\rm n}}/{\\cal B}\\sp{tp\\sp{\\rm n}+1}.$","abstract_has_math":true,"creators":["Benson, Steven Rex"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["McCulloh, Leon R.,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:26Z","date_published":"2014-12-16T06:18:26Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8908621"],"render_values":[{"text":"(UMI)AAI8908621","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71269","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["McCulloh, Leon R.,"]},{"key":"dc:creator","label":"Author","values":["Benson, Steven Rex"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:26Z","10000-01-01","1988"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71269","(UMI)AAI8908621"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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 $v\\sb{\\rm L}$ ($\\pi-\\pi\\sp\\prime$) $&gt;$ $v\\sb{\\rm L}(\\pi)$ (= $p\\sp{n}$), then $v\\sb{\\rm L}(\\pi-\\pi\\sp\\prime$) = $M\\sb{\\rm L}$(K,K$\\sp\\prime$).","More generally, if $t$ $&lt;$ $p$, 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$\\sp\\prime$)).","Made available in DSpace on 2014-12-16T06:18:26Z (GMT). No. of bitstreams: 1 8908621.pdf: 1257812 bytes, checksum: d0d3c16aefecc2e4bce00aad6851a0aa (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71435 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","50 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."]},{"key":"dc:title","label":"Title","values":["Canonical Invariants for Corresponding Residue Systems in P-Adic Fields"]}]}],"canonical_facts":{"dc:contributor":["McCulloh, Leon R.,"],"dc:creator":["Benson, Steven Rex"],"dc:date":["2014-12-16T06:18:26Z","10000-01-01","1988"],"dc:description":["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 $v\\sb{\\rm L}$ ($\\pi-\\pi\\sp\\prime$) $&gt;$ $v\\sb{\\rm L}(\\pi)$ (= $p\\sp{n}$), then $v\\sb{\\rm L}(\\pi-\\pi\\sp\\prime$) = $M\\sb{\\rm L}$(K,K$\\sp\\prime$).","More generally, if $t$ $&lt;$ $p$, 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$\\sp\\prime$)).","Made available in DSpace on 2014-12-16T06:18:26Z (GMT). No. of bitstreams: 1 8908621.pdf: 1257812 bytes, checksum: d0d3c16aefecc2e4bce00aad6851a0aa (MD5) Previous issue date: 1988","Embargo set by: Seth Robbins for item 71435 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","50 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1988."],"dc:identifier":["http://hdl.handle.net/2142/71269","(UMI)AAI8908621"],"dc:subject":["Mathematics"],"dc:title":["Canonical Invariants for Corresponding Residue Systems in P-Adic Fields"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}