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<span id="openzim-page-title" class="mw-page-title-main"><i>Local Fields</i></span>
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</style><div role="note" class="hatnote navigation-not-searchable">For The concept in mathematics, see <a href="Local_field" title="Local field">Local field</a>.</div>
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</style><table class="infobox ib-book vcard"><caption class="infobox-title">Local Fields </caption><tbody><tr><td colspan="2" class="infobox-image"></td></tr><tr><th scope="row" class="infobox-label">Author</th><td class="infobox-data"><a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a></td></tr><tr><th scope="row" class="infobox-label">Original title</th><td class="infobox-data"><i>Corps Locaux</i></td></tr><tr><th scope="row" class="infobox-label">Language</th><td class="infobox-data"><a href="French_language" title="French language">French</a> (original)<br> <a href="English_language" title="English language">English</a> (translation)</td></tr><tr><th scope="row" class="infobox-label">Subject</th><td class="infobox-data"><a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></td></tr><tr><th scope="row" class="infobox-label">Genre</th><td class="infobox-data">Non-fiction</td></tr><tr><th scope="row" class="infobox-label">Publisher</th><td class="infobox-data"><a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Publication date</div></th><td class="infobox-data">1980</td></tr><tr><th scope="row" class="infobox-label">Publication place</th><td class="infobox-data">France</td></tr><tr><th scope="row" class="infobox-label">Media type</th><td class="infobox-data">Print</td></tr><tr><th scope="row" class="infobox-label">Pages</th><td class="infobox-data">241 pp.</td></tr><tr><th scope="row" class="infobox-label"><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a></th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><bdi>978-0-387-90424-5</bdi></td></tr><tr><th scope="row" class="infobox-label"><a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)"><abbr title="Online Computer Library Center number">OCLC</abbr></a></th><td class="infobox-data"><a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/4933106">4933106</a></td></tr></tbody></table>
<p><i><b>Corps Locaux</b></i> by <a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a>, originally published in 1962 and translated into English as <i><b>Local Fields</b></i> by <a href="Marvin_Jay_Greenberg" class="mw-redirect" title="Marvin Jay Greenberg">Marvin Jay Greenberg</a> in 1979, is a seminal graduate-level <a href="Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a> text covering <a href="Local_fields" class="mw-redirect" title="Local fields">local fields</a>, <a href="Ramification_(mathematics)" title="Ramification (mathematics)">ramification</a>, <a href="Group_cohomology" title="Group cohomology">group cohomology</a>, and <a href="Local_class_field_theory" title="Local class field theory">local class field theory</a>. The book's end goal is to present local class field theory from the cohomological point of view.
</p><p>In this book, a "local field" is defined as a field <a href="Complete_metric_space" title="Complete metric space">complete</a> with respect to a <a href="Discrete_valuation" title="Discrete valuation">discrete valuation</a>, but current usage (including later works by Serre) add the condition that the <a href="Residue_field" title="Residue field">residue class field</a> is finite.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It has over 3500 citations in <a href="Google_Scholar" title="Google Scholar">Google Scholar</a>, and is often referenced with respect.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Contents">Contents</h2></div>
<ol><li><i>Part I, Local Fields (Basic Facts)</i>: Discrete valuation rings, Dedekind domains, and Completion.</li>
<li><i>Part II, Ramification</i>: Discriminant & Different, Ramification Groups, The Norm, and Artin Representation.</li>
<li><i>Part III, Group Cohomology</i>: Abelian & Nonabelian Cohomology, Cohomology of Finite Groups, Theorems of Tate and Nakayama, Galois Cohomology, Class Formations, and Computation of Cup Products.</li>
<li><i>Part IV, Local Class Field Theory</i>: Brauer Group of a Local Field, Local Class Field Theory, Local Symbols and Existence Theorem, and Ramification.</li></ol>
<div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFCassels1986" class="citation book cs1">Cassels, J. W. S. (1986). <i>Local Fields</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. v.</cite></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFBerg2020" class="citation web cs1">Berg, Michael (2020-02-23). <a rel="nofollow" class="external text" href="https://old.maa.org/press/maa-reviews/local-fields">"Local Fields"</a> <span class="cs1-format">(PDF)</span>. <i>MAA Reviews</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-05-30</span></span>.</cite></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRaskin2016" class="citation web cs1">Raskin, Sam (2016). <a rel="nofollow" class="external text" href="https://ocw.mit.edu/courses/18-786-number-theory-ii-class-field-theory-spring-2016/pages/readings/">"Number Theory II: Class Field Theory"</a>. <i>MIT OpenCourseWare Reviews</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2025-05-30</span></span>.</cite> <i>A classic reference that rewards the effort you put into it.</i></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/local+field+%28commutative+algebra%29">Local Fields</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a> <i>the famous text Corps Locaux by Serre</i></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFGuillot2018" class="citation book cs1">Guillot, Pierre (2018). <i>A gentle course in local class field theory: local number fields, Brauer groups, Galois cohomology</i>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</cite> <i>a classic, beautiful textbook</i></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><cite id="CITEREFSerre1980" class="citation cs2"><a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1980), <i>Local Fields</i>, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90424-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0554237">0554237</a></cite></li></ul>
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