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</style><div role="note" class="hatnote navigation-not-searchable">"Modern Algebra" redirects here. For the area of mathematics, see <a href="Modern_algebra" class="mw-redirect" title="Modern algebra">Modern algebra</a>.</div>

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</style><table class="infobox ib-book vcard"><caption class="infobox-title">Moderne Algebra </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="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Bartel Leendert van der Waerden</a></td></tr><tr><th scope="row" class="infobox-label">Language</th><td class="infobox-data">German</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">1930</td></tr></tbody></table>

<p><i><b>Moderne Algebra</b></i> is a two-volume German textbook on graduate <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a> by <a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">Bartel Leendert van der Waerden</a> (<a href="#CITEREFvan_der_Waerden1930">1930</a>, <a href="#CITEREFvan_der_Waerden1931">1931</a>), originally based on lectures given by <a href="Emil_Artin" title="Emil Artin">Emil Artin</a> in 1926 and by <a href="Emmy_Noether" title="Emmy Noether">Emmy Noether</a> (<a href="#CITEREFNoether1929">1929</a>) from 1924 to 1928. The English translation of 1949–1950 had the title <i><b>Modern algebra</b></i>, though a later, extensively revised edition in 1970 had the title <i><b>Algebra</b></i>.

</p><p>The book was one of the first <a href="Textbook" title="Textbook">textbooks</a> to use an abstract axiomatic approach to <a href="Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>, and <a href="Field_(mathematics)" title="Field (mathematics)">fields</a>, and was by far the most successful, becoming the standard reference for <a href="Graduate_school" class="mw-redirect" title="Graduate school">graduate</a> algebra for several decades. It "had a tremendous impact, and is widely considered to be the major text on algebra in the twentieth century."<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>In 1975 van der Waerden described the sources he drew upon to write the book.<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>

</p><p>In 1997 <a href="Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a> recollected the book's influence:<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>

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<ul><li>Upon its publication it was soon clear that this was the way that algebra should be presented.</li>

<li>Its simple but austere style set the pattern for mathematical texts in other subjects, from <a href="Banach_algebra" title="Banach algebra">Banach algebras</a> to <a href="Topological_group" title="Topological group">topological group</a> theory.</li>

<li>[Van der Waerden's] two volumes on modern algebra ... dramatically changed the way algebra is now taught by providing a decisive example of a clear and perspicacious presentation. It is, in my view, the most influential text of algebra of the twentieth century.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Publication_history">Publication history</h2></div>

<p><i><b>Moderne Algebra</b></i> has a rather confusing publication history, because it went through many different editions, several of which were extensively rewritten with chapters and major topics added, deleted, or rearranged. In addition the new editions of first and second volumes were issued almost independently and at different times, and the numbering of the English editions does not correspond to the numbering of the German editions. In 1955 the title was changed from "Moderne Algebra" to "Algebra" following a suggestion of Brandt, with the result that the two volumes of the third German edition do not even have the same title.

</p><p>For volume 1, the first German edition was published in 1930, the second in 1937 (with the axiom of choice removed), the third in 1951 (with the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> reinstated, and with more on <a href="Valuation_(algebra)" title="Valuation (algebra)">valuations</a>).<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> The fourth edition appeared in 1955 (with the title changed to <i>Algebra</i>), the fifth in 1960, the sixth in 1964, the seventh in 1966, the eighth in 1971, the ninth in 1993. For volume 2, the first edition was published in 1931, the second in 1940, the third in 1955 (with the title changed to <i>Algebra</i>), the fourth in 1959 (extensively rewritten, with elimination theory replaced by algebraic functions of 1 variable),<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> the fifth in 1967, and the sixth in 1993. The German editions were all published by Springer.

</p><p>The first English edition was published in 1949–1950 and was a translation of the second German edition. There was a second edition in 1953, and a third edition under the new title <b>Algebra</b> in 1970 translated from the 7th German edition of volume 1 and the 5th German edition of volume 2. The three English editions were originally published by Ungar, though the 3rd English edition was later reprinted by Springer.

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</style><cite id="CITEREFvan_der_Waerden1930" class="citation book cs1 cs1-prop-foreign-lang-source">van der Waerden, Bartel Leendert (1930). <i>Moderne Algebra. Teil I</i>. Die Grundlehren der mathematischen Wissenschaften (in German). Vol. 33. 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>978-3-540-56799-8</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=0009016">0009016</a>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li>

<li><cite id="CITEREFvan_der_Waerden1931" class="citation book cs1 cs1-prop-foreign-lang-source">van der Waerden, Bartel Leendert (1931). <i>Moderne Algebra. Teil II</i>. Die Grundlehren der mathematischen Wissenschaften (in German). Vol. 34. Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-56801-8</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=0002841">0002841</a>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></li>

<li><cite id="CITEREFvan_der_Waerden1949" class="citation book cs1">van der Waerden, Bartel Leendert (1949). <i>Modern Algebra. Vol I</i>. Translated by Blum, Fred. New York, N. Y.: Frederick Ungar Publishing Co. <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=0029363">0029363</a>.</cite></li>

<li><cite id="CITEREFvan_der_Waerden1950" class="citation book cs1">van der Waerden, Bartel Leendert (1950). <i>Modern Algebra. Vol II</i>. Translated by Blum, Fred. New York, N. Y.: Frederick Ungar Publishing Co.</cite></li>

<li><cite id="CITEREFvan_der_Waerden1970" class="citation book cs1">van der Waerden, Bartel Leendert (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xw8lAQAAIAAJ"><i>Algebra. Vol 1</i></a>. Translated by Blum, Fred; Schulenberger, John R. New York, N. Y.: Frederick Ungar Publishing Co. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-40624-4</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=0263582">0263582</a>. Reprinted by Springer.</cite></li>

<li><cite id="CITEREFvan_der_Waerden1970" class="citation book cs1">van der Waerden, Bartel Leendert (1970). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oWzoOWp_tWgC"><i>Algebra. Vol. 2</i></a>. Translated by Schulenberger, John R. (3rd ed.). New York, N. Y.: Frederick Ungar Publishing Co. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780387406251</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=0263583">0263583</a>. Reprinted by Springer.</cite></li></ul>

<p>There were also Russian editions published in 1976 and 1979, and Japanese editions published in 1959 and 1967–1971.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Bernard Behan & <a href="Karl_Sigmund" title="Karl Sigmund">Karl Sigmund</a> (2008) "A Short Tale of Two Cities: Otto Schrier and the Hamburg–Vienna Connection", p. 33, <a href="Mathematical_Intelligencer" class="mw-redirect" title="Mathematical Intelligencer">Mathematical Intelligencer</a> 30(3):27–35.</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="CITEREFvan_der_Waerden1975" class="citation cs2"><a href="Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, Bartel Leendert</a> (1975), "On the sources of my book Moderne Algebra", <i><a href="Historia_Mathematica" title="Historia Mathematica">Historia Mathematica</a></i>, <b>2</b>: <span class="nowrap">31–</span>40, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0315-0860%2875%2990034-8">10.1016/0315-0860(75)90034-8</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0315-0860">0315-0860</a>, <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=0465721">0465721</a></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="CITEREFMac_Lane1997" class="citation cs2"><a href="Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1997), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/199703/maclane.pdf">"Van der Waerden's Modern algebra"</a> <span class="cs1-format">(PDF)</span>, <i><a href="Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>, <b>44</b> (3): <span class="nowrap">321–</span>322, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9920">0002-9920</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230325171728/https://www.ams.org/notices/199703/maclane.pdf">archived</a> <span class="cs1-format">(PDF)</span> from the original on 2023-03-25<span class="reference-accessdate">, retrieved <span class="nowrap">2023-12-03</span></span></cite></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"><cite id="CITEREFZelinsky1951" class="citation cs2"><a href="Daniel_Zelinsky_(mathematician)" title="Daniel Zelinsky (mathematician)">Zelinsky, Daniel</a> (1951), "Book Review: Moderne Algebra", <i><a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, <b>57</b> (3): <span class="nowrap">206–</span>207, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1951-09496-3">10.1090/S0002-9904-1951-09496-3</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9904">0002-9904</a></cite></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="CITEREFZelinsky,_Daniel1961" class="citation journal cs1">Zelinsky, Daniel (1961). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1961-10504-1">"Review of <i>Algebra, Part 2 (4th ed. of Moderne Algebra)</i> by B. L. van der Waerden"</a>. <i>Bull. Amer. Math. Soc</i>. <b>67</b>: <span class="nowrap">83–</span>86. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1961-10504-1">10.1090/S0002-9904-1961-10504-1</a></span>.</cite></span>

</li>

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<ul><li><cite id="CITEREFHahnTaussky1932" class="citation cs2"><a href="Hans_Hahn_(mathematician)" title="Hans Hahn (mathematician)">Hahn, H.</a>; <a href="Olga_Taussky-Todd" title="Olga Taussky-Todd">Taussky, Olga</a> (1932), "Literaturberichte: Moderne Algebra", <i>Monatshefte für Mathematik und Physik</i>, <b>39</b> (1): <span class="nowrap">A11 –</span> <span class="nowrap">A12</span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01699101">10.1007/BF01699101</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-9255">0026-9255</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:197662655">197662655</a></cite></li>

<li><cite id="CITEREFHofreiter1936" class="citation cs2">Hofreiter (1936), "Literaturberichte: Moderne Algebra", <i>Monatshefte für Mathematik und Physik</i>, <b>45</b> (1): <span class="nowrap">A25 –</span> <span class="nowrap">A26</span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01708057">10.1007/BF01708057</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-9255">0026-9255</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124462638">124462638</a></cite></li>

<li><cite id="CITEREFHofreiter1941" class="citation cs2">Hofreiter (1941), "Literaturberichte: Moderne Algebra", <i>Monatshefte für Mathematik und Physik</i>, <b>49</b> (1): A21, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01707351">10.1007/BF01707351</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-9255">0026-9255</a></cite></li>

<li><cite id="CITEREFOre1931" class="citation cs2">Ore, Oystein (1931), "Recent Publications: Reviews: Moderne Algebra", <i><a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">The American Mathematical Monthly</a></i>, <b>38</b> (4): 226, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2301771">10.2307/2301771</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2301771">2301771</a></cite></li>

<li><cite id="CITEREFOre1932" class="citation cs2">Ore, Oystein (1932), "Book Review: Moderne Algebra", <i><a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, <b>38</b> (5): <span class="nowrap">327–</span>329, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1932-05385-X">10.1090/S0002-9904-1932-05385-X</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9904">0002-9904</a></cite></li>

<li><cite id="CITEREFOre1938" class="citation cs2">Ore, Oystein (1938), "Book Review: Moderne Algebra", <i><a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, <b>44</b> (5): <span class="nowrap">320–</span>321, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1938-06737-7">10.1090/S0002-9904-1938-06737-7</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9904">0002-9904</a></cite></li>

<li><cite id="CITEREFNoether1929" class="citation cs2"><a href="Emmy_Noether" title="Emmy Noether">Noether, Emmy</a> (1929), "Hyperkomplexe Größen und Darstellungstheorie", <i><a href="Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a></i>, <b>30</b>: <span class="nowrap">641–</span>692, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01187794">10.1007/BF01187794</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5874">0025-5874</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120464373">120464373</a></cite></li>

<li><cite id="CITEREFSchlote2005" class="citation cs2">Schlote, K.-H. (2005), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=UdGBy8iLpocC&pg=PA901">"Chapter 70 B. L. van der Waerden, Moderne Algebra"</a>, in Grattan-Guinness, Ivor (ed.), <i>Landmark writings in western mathematics 1640--1940</i>, Elsevier B. V., Amsterdam, pp. <span class="nowrap">901–</span>916, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-50871-3</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=2169816">2169816</a></cite></li>

<li><cite id="CITEREFTaussky1933" class="citation cs2"><a href="Olga_Taussky-Todd" title="Olga Taussky-Todd">Taussky, Olga</a> (1933), "Literaturberichte: Moderne Algebra", <i>Monatshefte für Mathematik und Physik</i>, <b>40</b> (1): <span class="nowrap">A3 –</span> <span class="nowrap">A4</span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01708891">10.1007/BF01708891</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0026-9255">0026-9255</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124038027">124038027</a></cite></li></ul></div><div><div class="zim-footer">

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