{"id":{"repo_id":"durham","oai_identifier":"oai:etheses.durham.ac.uk:471"},"canonical_url":"https://search.dev.ndltd.org/etd/durham/oai:etheses.durham.ac.uk:471","repository":{"repo_id":"durham","name":"Durham University","base_url":"http://etheses.dur.ac.uk/cgi/oai2"},"display":{"title":"The Field of Norms Functor and the Hilbert Symbol","abstract":"The classical Hilbert symbol of a higher local field $F$ containing a primitive $p^M$-th root of unity $\\zeta_M$ is a pairing $F^*/(F^*)^{p^M}\\times K_N(F)/p^M \\to \\mu_{p^M}$, describing Kummer extensions of exponent $p^M$. In this thesis we define a generalised Hilbert symbol and prove a formula for it. Our approach has several ingredients. The field of norms functor of Scholl associates to any strictly deeply ramified tower $F_.$ a field $\\c F$ of characteristic $p$. Separable extensions of $\\cal F$ correspond functorially to extensions of $F_.$, giving rise to $\\Gamma_{\\cal F}\\cong \\Gamma_{F_{\\infty}}\\subset \\Gamma_F$. We define morphisms $\\cal N_{\\cal F/F_n}: K_N^t(\\cal F)/p^M \\to K_N^t(F_n)/p^M$ which are compatible with the norms $N_{F_{n+m}/F_n}$ for every $m$. Using these, we show that field of norms functor commutes with the reciprocity maps $\\Psi_{\\cal F}: K_N^t(\\cal F) \\to \\Gamma_{\\cal F}^{ab}$ and $\\Psi_{F_n}: K_N^t(F_n) \\to \\Gamma_{F_n}^{ab}$ constructed by Fesenko. Imitating Fontaine's approach, we obtain an invariant form of Parshin's formula for the Witt pairing in characteristic $p$. The `main lemma' relates Kummer extensions of $F$ and Witt extensions of $\\cal F$, allowing us to derive a formula for the generalised Hilbert symbol $\\hat F_{\\infty}^* \\times K_N(\\cal F) \\to \\mu_{p^M}$, where $\\hat F_{\\infty}$ is the $p$-adic completion of $\\varinjlim_n F_n$.","abstract_html":"The classical Hilbert symbol of a higher local field $F$ containing a primitive <span class=\"etd-inline-math\">p<sup>M</sup></span>-th root of unity <span class=\"etd-inline-math\">\\zeta<sub>M</sub></span> is a pairing <span class=\"etd-inline-math\">F<sup>*</sup>/(F<sup>*</sup>)<sup>p<sup>M</sup></sup>\\times K<sub>N</sub>(F)/p<sup>M</sup> \\to &mu;<sub>p<sup>M</sup></sub></span>, describing Kummer extensions of exponent <span class=\"etd-inline-math\">p<sup>M</sup></span>. In this thesis we define a generalised Hilbert symbol and prove a formula for it. Our approach has several ingredients. The field of norms functor of Scholl associates to any strictly deeply ramified tower <span class=\"etd-inline-math\">F<sub>.</sub></span> a field $\\c F$ of characteristic $p$. Separable extensions of $\\cal F$ correspond functorially to extensions of <span class=\"etd-inline-math\">F<sub>.</sub></span>, giving rise to <span class=\"etd-inline-math\">\\Gamma<sub>\\cal F</sub>\\cong \\Gamma<sub>F<sub>\\infty</sub></sub>\\subset \\Gamma<sub>F</sub></span>. We define morphisms <span class=\"etd-inline-math\">\\cal N<sub>\\cal F/F<sub>n</sub></sub>: K<sub>N</sub><sup>t</sup>(\\cal F)/p<sup>M</sup> \\to K<sub>N</sub><sup>t</sup>(F<sub>n</sub>)/p<sup>M</sup></span> which are compatible with the norms <span class=\"etd-inline-math\">N<sub>F<sub>n+m</sub>/F<sub>n</sub></sub></span> for every $m$. Using these, we show that field of norms functor commutes with the reciprocity maps <span class=\"etd-inline-math\">\\Psi<sub>\\cal F</sub>: K<sub>N</sub><sup>t</sup>(\\cal F) \\to \\Gamma<sub>\\cal F</sub><sup>ab</sup></span> and <span class=\"etd-inline-math\">\\Psi<sub>F<sub>n</sub></sub>: K<sub>N</sub><sup>t</sup>(F<sub>n</sub>) \\to \\Gamma<sub>F<sub>n</sub></sub><sup>ab</sup></span> constructed by Fesenko. Imitating Fontaine&#x27;s approach, we obtain an invariant form of Parshin&#x27;s formula for the Witt pairing in characteristic $p$. The `main lemma&#x27; relates Kummer extensions of $F$ and Witt extensions of $\\cal F$, allowing us to derive a formula for the generalised Hilbert symbol <span class=\"etd-inline-math\">\\hat F<sub>\\infty</sub><sup>*</sup> \\times K<sub>N</sub>(\\cal F) \\to &mu;<sub>p<sup>M</sup></sub></span>, where <span class=\"etd-inline-math\">\\hat F<sub>\\infty</sub></span> is the $p$-adic completion of <span class=\"etd-inline-math\">\\varinjlim<sub>n</sub> F<sub>n</sub></span>.","abstract_has_math":true,"creators":["Jenni, Ruth Christine"],"institution":"Durham University","degree_name":"PhD","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-24T02:11:09Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Jenni, Ruth Christine"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2010"]},{"key":"dc:date.issued","label":"Date","values":["2010"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Mathematical Sciences, Department of"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["Durham University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://etheses.durham.ac.uk/id/eprint/471/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["PhD"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://etheses.durham.ac.uk/id/eprint/471/1/RCJenni.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The classical Hilbert symbol of a higher local field $F$ containing a primitive $p^M$-th root of unity $\\zeta_M$ is a pairing $F^*/(F^*)^{p^M}\\times K_N(F)/p^M \\to \\mu_{p^M}$, describing Kummer extensions of exponent $p^M$. In this thesis we define a generalised Hilbert symbol and prove a formula for it. Our approach has several ingredients. The field of norms functor of Scholl associates to any strictly deeply ramified tower $F_.$ a field $\\c F$ of characteristic $p$. Separable extensions of $\\cal F$ correspond functorially to extensions of $F_.$, giving rise to $\\Gamma_{\\cal F}\\cong \\Gamma_{F_{\\infty}}\\subset \\Gamma_F$. We define morphisms $\\cal N_{\\cal F/F_n}: K_N^t(\\cal F)/p^M \\to K_N^t(F_n)/p^M$ which are compatible with the norms $N_{F_{n+m}/F_n}$ for every $m$. Using these, we show that field of norms functor commutes with the reciprocity maps $\\Psi_{\\cal F}: K_N^t(\\cal F) \\to \\Gamma_{\\cal F}^{ab}$ and $\\Psi_{F_n}: K_N^t(F_n) \\to \\Gamma_{F_n}^{ab}$ constructed by Fesenko. Imitating Fontaine's approach, we obtain an invariant form of Parshin's formula for the Witt pairing in characteristic $p$. The `main lemma' relates Kummer extensions of $F$ and Witt extensions of $\\cal F$, allowing us to derive a formula for the generalised Hilbert symbol $\\hat F_{\\infty}^* \\times K_N(\\cal F) \\to \\mu_{p^M}$, where $\\hat F_{\\infty}$ is the $p$-adic completion of $\\varinjlim_n F_n$."]},{"key":"dc:format","label":"Dc Format","values":["text"]},{"key":"dc:title","label":"Title","values":["The Field of Norms Functor and the Hilbert Symbol"]}]}],"canonical_facts":{"dc:creator":["Jenni, Ruth Christine"],"dc:date":["2010"],"dc:date.issued":["2010"],"dc:description.abstract":["The classical Hilbert symbol of a higher local field $F$ containing a primitive $p^M$-th root of unity $\\zeta_M$ is a pairing $F^*/(F^*)^{p^M}\\times K_N(F)/p^M \\to \\mu_{p^M}$, describing Kummer extensions of exponent $p^M$. In this thesis we define a generalised Hilbert symbol and prove a formula for it. Our approach has several ingredients. The field of norms functor of Scholl associates to any strictly deeply ramified tower $F_.$ a field $\\c F$ of characteristic $p$. Separable extensions of $\\cal F$ correspond functorially to extensions of $F_.$, giving rise to $\\Gamma_{\\cal F}\\cong \\Gamma_{F_{\\infty}}\\subset \\Gamma_F$. We define morphisms $\\cal N_{\\cal F/F_n}: K_N^t(\\cal F)/p^M \\to K_N^t(F_n)/p^M$ which are compatible with the norms $N_{F_{n+m}/F_n}$ for every $m$. Using these, we show that field of norms functor commutes with the reciprocity maps $\\Psi_{\\cal F}: K_N^t(\\cal F) \\to \\Gamma_{\\cal F}^{ab}$ and $\\Psi_{F_n}: K_N^t(F_n) \\to \\Gamma_{F_n}^{ab}$ constructed by Fesenko. Imitating Fontaine's approach, we obtain an invariant form of Parshin's formula for the Witt pairing in characteristic $p$. The `main lemma' relates Kummer extensions of $F$ and Witt extensions of $\\cal F$, allowing us to derive a formula for the generalised Hilbert symbol $\\hat F_{\\infty}^* \\times K_N(\\cal F) \\to \\mu_{p^M}$, where $\\hat F_{\\infty}$ is the $p$-adic completion of $\\varinjlim_n F_n$."],"dc:format":["text"],"dc:identifier.uri":["https://etheses.durham.ac.uk/id/eprint/471/1/RCJenni.pdf"],"dc:publisher.department":["Mathematical Sciences, Department of"],"dc:publisher.institution":["Durham University"],"dc:relation.isreferencedby":["https://etheses.durham.ac.uk/id/eprint/471/"],"dc:title":["The Field of Norms Functor and the Hilbert Symbol"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["PhD"]},"updated_at":"2026-07-24T02:11:09Z"}