Document Zbl 0193.34701

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(1965). </div> <div class="abstract">Das Buch besteht aus den drei Teilen: “Gruppen, Ringe und Moduln”, “K枚rpertheorie” und “Lineare Algebra und Darstellungstheorie”, welche ihrerseits in Kapitel unterteilt sind.<br class="zbmathjax-paragraph">Schon im ersten Teil wird die Tendenz des Autors deutlich, nicht eine systematische Einf眉hrung von Begriffen samt Aufz盲hlung mehr oder weniger einfacher Eigenschaften anzustreben, sondern m枚glichst bald relevante Aussagen herzuleiten. Die Kapitel眉berschriften von Teil I lauten: “Gruppen”, “Ringe”, “Moduln”, “Homologie”, “Polynome” und “Noethersche Ringe und Moduln”. Nach kuzer Behandlung des Gruppenbegriffs, welche S盲tze 眉ber Sylow-Gruppen einschlie脽t, werden im ersten Kapitel Kategorien, Funktoren und kategorientheoretische Konstruktionen eingef眉hrt. Es folgen Anwendungen dieser Begriffe bei verschiedenen universellen Konstruktionen. Im Kapitel 眉ber Ringe findet man den Chinesischen Restsatz und das Lokalisierungsprinzip. Das vierte Kapitel behandelt auf wenigen Seiten die Homologiesequenz von Modulkomplexen, die Eulersche Charakteristik eines Komplexes und den Satz von Jordan-H枚lder. Das letzte Kapitel schlie脽t mit der Herleitung der Prim盲rzerlegung f眉r Noethersche Moduln.<br class="zbmathjax-paragraph">Teil II umfa脽t die Kapitel “Algebraische Erweiterungen”, “Galois-Theorie”, “Ringerweiterungen”, “Transzendente Erweiterungen”, “Reelle K枚rper”, “Bewertungen”. Grundbegriffe der K枚rpertheorie enth盲lt das erste Kapitel. Das n盲chste Kapitel lehnt sich im Aufbau an Artins bekannte “Galoissche Theorie” [<span class="zbmathjax-textit">E. Artin</span>, Galoissche Theorie (Leipzig 1959, dies. <a href="/0086.25703">Zbl 0086.25703</a>; 2. Auflage 1965; <a href="/0133.26402">Zbl 0133.26402</a>)] an. Es folgen Betrachtungen 眉ber ganz-abgeschlossene Ringe; in galoisschen Erweiterungen derselben werden die auf Hilbert zur眉ckgehenden Begriffe Zerlegungsgruppe und Tr盲gheitsgruppe entwickelt. Im Kapitel 眉ber transzendente Erweiterungen findet sich neben Hilberts Nullstellensatz und Bemerkungen 眉ber algebraische Mannigfaltigkeiten ein Abschnitt 眉ber Derivationen mit Anwendungen auf separabel erzeugte K枚rpererweiterungen. Die Artin-Schreier Theorie reeller K枚rper wird im n盲chsten Kapitel behandelt. Grundbegriffe der Bewertungstheorie sind im letzten Kapitel zusammengestellt.<br class="zbmathjax-paragraph">Mit einem Kapitel 眉ber Matrizen und lineare Abbildungen beginnt Teil III. Der Struktur von Bilinearformen ist das folgende Kapitel gewidmet. In diesem Zusammenhang wird der Satz von Witt bewiesen, die Wittsche Gruppe und die Witt-Grothendieck-Gruppe eingef眉hrt, Der Sylvestersche Tr盲gheitssatz behandelt und die zu einer symmetrischen Bilinearform geh枚rige Clifford-Algebra konstruiert. Abschlie脽end wird der Spektralsatz f眉r den Hermiteschen und den symmetrischen Fall bewiesen. Strukturuntersuchungen von Moduln 眉ber Hauptidealbereichen f眉hren zur Herleitung der Jordanschen Normalform einer Matrix. Ein weiteres Kapitel behandelt Tensorprodukte, alternierende und symmetrische Produkte und den Grothendieck-Ring einer Gruppe. Die Strukturtheorie halbeinfacher Ringe wird aus dem Dichtesatz von Jacobson entwickelt. Ein Kapitel 眉ber Darstellungstheorie endlicher Gruppen beschlie脽t diesen dritten Teil, dessen Leitthema die Struktur von Moduln und Ringen ist. Ein Anhang enth盲lt Transzendenzbeweise f眉r $e$ und $\pi$. An jedes Kapitel schlie脽t eine Reihe von meist schwierigen 脺bungsaufgaben, die den im Textteil dargestellten Stoff erg盲nzen.<br class="zbmathjax-paragraph">Einem Leser ohne algebraische Vorkenntnisse und Routine wird das Buch vermutlich Schwierigkeiten bereiten. Daher sollte man es dem H枚rer eines Kurses 眉ber Algebra h枚chstens als begleitende Lekt眉re in die Hand geben; im Anschlu脽 an eine Vorlesung sollte man es aber jedem besonders empfehlen. Nicht wie bei Bourbaki ist eine m枚glichst systematische Darstellung das Ziel des Autors. Er w盲hlt vielmehr aus und setzt Schwerpunkte. Dies gelingt so vortrefflich, da脽 man in lebendiger Darstellungsweise auf knappem Raum eine imponierende F眉lle an Material zusammengetragen findet. Der Leser wird in die verschiedensten Bereiche der Algebra eingef眉hrt. Da die einzelnen Kapitel weitgehend unabh盲ngig voneinander lesbar sind, eignet es sich vorz眉glich zur Orientierung. Seit B. L. van der Waerdens “Algebra” d眉rfte hier neben Bourbakis Werk das umfassendste Lehrbuch der Algebra in zeitgem盲脽er Sprache vorliegen, das sich in den Jahren nach seinem Erscheinen herforragend bew盲hrt und einen festen Platz in der Literatur erworben hat.<div class="reviewer"> Reviewer: <a href="/authors/?q=K%2A+Plewe">K. Plewe</a></div> <div class="clearfix"></div><div class="scan"> <span> Page: <a class="goto" href="/scans/193/342.gif">−5</a> <a class="goto" href="/scans/193/343.gif">−4</a> <a class="goto" href="/scans/193/344.gif">−3</a> <a class="goto" href="/scans/193/345.gif">−2</a> <a class="goto" href="/scans/193/346.gif">−1</a> <a class="goto" href="/scans/193/347.gif">±0</a> <a class="goto" href="/scans/193/348.gif">+1</a> <a class="goto" href="/scans/193/349.gif">+2</a> <a class="goto" href="/scans/193/350.gif">+3</a> <a class="goto" href="/scans/193/351.gif">+4</a> <a class="goto" href="/scans/193/352.gif">+5</a></span> <a class="scan" href="/scans/193/347.gif">Show Scanned Page</a><br> <img src="/scans/193/347.gif" alt="scan of Zbl 0193.34701" onclick="$(this).siblings('.scan').click();"> </div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3307642">Cited in <strong>9</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A3307642">Cited in <strong>465</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A00A05" title="MSC2020">00A05</a> </td> <td class="space"> Mathematics in general </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A12-01" title="MSC2020">12-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A13-01" title="MSC2020">13-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15-01" title="MSC2020">15-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A16-01" title="MSC2020">16-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A18-01" title="MSC2020">18-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20-01" title="MSC2020">20-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acommutative+algebra">commutative algebra</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0086.25703">Zbl 0086.25703</a>; <a href="/0133.26402">Zbl 0133.26402</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0193.34701" data-ciurl="" data-biburl="/bibtex/03307642.bib" data-amsurl="/amsrefs/03307642.bib" data-xmlurl="/xml/03307642.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03307642.pdf" title="Zbl 0193.34701 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2024 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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