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<!DOCTYPE html><html><head><title>A Field Guide to Algebra - Antoine Chambert-Loir - Google Books</title><link rel="stylesheet" href="/books/css/_a33f2a89320471e58c940b9287b9d4eb/kl_about_this_book_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/atb_a33f2a89320471e58c940b9287b9d4eb__en.js"></script><link rel="canonical" href="https://books.google.com/books/about/A_Field_Guide_to_Algebra.html?id=Rz2VlLa3paEC"/><meta property="og:url" content="https://books.google.com/books/about/A_Field_Guide_to_Algebra.html?id=Rz2VlLa3paEC"/><meta name="title" content="A Field Guide to Algebra"/><meta name="description" content="This is a small book on algebra where the stress is laid on the structure of ?elds, hence its title. Youwillhearaboutequations,bothpolynomialanddi?erential,andabout the algebraic structure of their solutions. For example, it has been known for centuries how to explicitely solve polynomial equations of degree 2 (Baby- nians, many centuries ago), 3 (Scipione del Ferro, Tartaglia, Cardan, around th 1500a.d.), and even 4 (Cardan, Ferrari,xvi century), using only algebraic operations and radicals (nth roots). However, the case of degree 5 remained unsolved until Abel showed in 1826 that a general equation of degree 5 cannot be solved that way. Soon after that, Galois de?ned the group of a polynomial equation as the group of permutations of its roots (say, complex roots) that preserve all algebraicidentitieswithrationalcoe?cientssatis?edbytheseroots.Examples of such identities are given by the elementary symmetric polynomials, for it is well known that the coe?cients of a polynomial are (up to sign) elementary symmetric polynomials in the roots. In general, all relations are obtained by combining these, but sometimes there are new ones and the group of the equation is smaller than the whole permutation group. Galois understood how this symmetry group can be used to characterize the solvability of the equation. He de?ned the notion of solvable group and showed that if the group of the equation is solvable, then one can express its roots with radicals, and conversely."/><meta property="og:title" content="A Field Guide to Algebra"/><meta property="og:type" content="book"/><meta property="og:site_name" content="Google Books"/><meta property="og:image" content="https://books.google.com.sg/books/content?id=Rz2VlLa3paEC&printsec=frontcover&img=1&zoom=1&imgtk=AFLRE737j4q2fDgSHVmFywWjPmTq7GH05pXAv7fiXwmsedb6shGibbt3xw1P88GsPMBdo0IefEk1M3FEhjUnq8mxivtvUN2wof4ZBvY-Y8cIKEFhtAsmdZu-oFzvz6m90lHnv006ATfT"/><link rel="image_src" href="https://books.google.com.sg/books/content?id=Rz2VlLa3paEC&printsec=frontcover&img=1&zoom=1&imgtk=AFLRE737j4q2fDgSHVmFywWjPmTq7GH05pXAv7fiXwmsedb6shGibbt3xw1P88GsPMBdo0IefEk1M3FEhjUnq8mxivtvUN2wof4ZBvY-Y8cIKEFhtAsmdZu-oFzvz6m90lHnv006ATfT"/><script></script><style>#gbar,#guser{font-size:13px;padding-top:1px !important;}#gbar{height:22px}#guser{padding-bottom:7px !important;text-align:right}.gbh,.gbd{border-top:1px solid #c9d7f1;font-size:1px}.gbh{height:0;position:absolute;top:24px;width:100%}@media all{.gb1{height:22px;margin-right:.5em;vertical-align:top}#gbar{float:left}}a.gb1,a.gb4{text-decoration:underline !important}a.gb1,a.gb4{color:#00c !important}.gbi .gb4{color:#dd8e27 !important}.gbf .gb4{color:#900 !important} #gbar { padding:.3em .6em !important;}</style></head><body class=""><div id=gbar><nobr><a target=_blank class=gb1 href="https://www.google.com.sg/search?tab=pw">Search</a> <a target=_blank class=gb1 href="https://www.google.com.sg/imghp?hl=en&tab=pi">Images</a> <a target=_blank class=gb1 href="https://maps.google.com.sg/maps?hl=en&tab=pl">Maps</a> <a target=_blank class=gb1 href="https://play.google.com/?hl=en&tab=p8">Play</a> <a target=_blank class=gb1 href="https://www.youtube.com/?tab=p1">YouTube</a> <a target=_blank class=gb1 href="https://news.google.com/?tab=pn">News</a> <a target=_blank class=gb1 href="https://mail.google.com/mail/?tab=pm">Gmail</a> <a target=_blank class=gb1 href="https://drive.google.com/?tab=po">Drive</a> <a target=_blank class=gb1 style="text-decoration:none" href="https://www.google.com.sg/intl/en/about/products?tab=ph"><u>More</u> »</a></nobr></div><div id=guser width=100%><nobr><span id=gbn class=gbi></span><span id=gbf class=gbf></span><span id=gbe></span><a target=_top id=gb_70 href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fid%3DRz2VlLa3paEC%26dq%3Dbibliogroup:%2522Undergraduate%2BTexts%2Bin%2BMathematics%2522%26hl%3Den%26sa%3DX%26redir_esc%3Dy&hl=en&ec=GAZACg" class=gb4>Sign in</a></nobr></div><div class=gbh style=left:0></div><div class=gbh style=right:0></div><div role="alert" style="position: absolute; left: 0; right: 0;"><a href="https://books.google.com.sg/books?id=Rz2VlLa3paEC&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&hl=en&sa=X&redir_esc=y&output=html_text" title="Screen reader users: click this link for accessible mode. 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Youwillhearaboutequations,bothpolynomialanddi?erential,andabout the algebraic structure of their solutions. For example, it has been known for centuries how to explicitely solve polynomial equations of degree 2 (Baby- nians, many centuries ago), 3 (Scipione del Ferro, Tartaglia, Cardan, around th 1500a.d.), and even 4 (Cardan, Ferrari,xvi century), using only algebraic operations and radicals (nth roots). However, the case of degree 5 remained unsolved until Abel showed in 1826 that a general equation of degree 5 cannot be solved that way. Soon after that, Galois de?ned the group of a polynomial equation as the group of permutations of its roots (say, complex roots) that preserve all algebraicidentitieswithrationalcoe?cientssatis?edbytheseroots.Examples of such identities are given by the elementary symmetric polynomials, for it is well known that the coe?cients of a polynomial are (up to sign) elementary symmetric polynomials in the roots. In general, all relations are obtained by combining these, but sometimes there are new ones and the group of the equation is smaller than the whole permutation group. Galois understood how this symmetry group can be used to characterize the solvability of the equation. He de?ned the notion of solvable group and showed that if the group of the equation is solvable, then one can express its roots with radicals, and conversely.</div></div></div><div class="search_box_wrapper"></div></td> </tr></table><div id="summary-second-column"></div></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="book_other_versions_anchor"></a>Other editions - <a href='https://books.google.com.sg/books?q=editions:ISBN0387214283&id=Rz2VlLa3paEC'>View all</a></h3><div id=book_other_versions class=about_content><div id=book_other_versions_v><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=J53qAlCYoG4C&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=1" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=J53qAlCYoG4C&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=J53qAlCYoG4C&printsec=frontcover&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=1"><span dir=ltr>A Field Guide to Algebra</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Antoine+Chambert-Loir%22" class="secondary"><span dir=ltr>Antoine Chambert-Loir</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2004</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=PDJ62DSHSPQC&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=1" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=PDJ62DSHSPQC&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=PDJ62DSHSPQC&printsec=frontcover&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=1"><span dir=ltr>A Field Guide to Algebra</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Antoine+Chambert-Loir%22" class="secondary"><span dir=ltr>Antoine Chambert-Loir</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2007</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=qA6hcQAACAAJ&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=1" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=qA6hcQAACAAJ&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=qA6hcQAACAAJ&dq=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_book_other_versions_r&cad=1"><span dir=ltr>A Field Guide to Algebra</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Antoine+Chambert-Loir%22" class="secondary"><span dir=ltr>Antoine Chambert-Loir</span></a></span><br/><span><span style="color:#999">No preview available</span> - 2010</span><br/></span></div></td><td align=right></td></tr></table></div></div><script>(function () {var fn = window['_OC_WSBookList'] || window['_OC_BookList'];fn && fn('book_other_versions', [{"title":"A Field Guide to Algebra","authors":"Antoine Chambert-Loir","bib_key":"ISBN:9780387214283","pub_date":"1 Oct 2004","snippet":"This book has a nonstandard choice of topics, including material on differential galois groups and proofs of the transcendence of e and pi. 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font-weight:normal">(2004)</span></h3><div id=about_author class=about_content><div id=about_author_v><div class=textmodulecontent><p>Antoine Chambert-Loir聽is Professor at Universit茅 de Rennes 1.</p></div></div></div></div><div class=vertical_module_list_row><h3 class="about_title">Bibliographic information</h3><div class="about_content" id="metadata_content" style="padding-bottom:.3em"><div class="metadata_sectionwrap"><table id="metadata_content_table"><tr class="metadata_row"><td class="metadata_label">Title</td><td class="metadata_value"><span dir=ltr>A Field Guide to Algebra</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_metadata_r&cad=3"><i><span dir=ltr>Undergraduate Texts in Mathematics</span></i></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Author</span></td><td class="metadata_value"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22Antoine+Chambert-Loir%22&source=gbs_metadata_r&cad=3"><span dir=ltr>Antoine Chambert-Loir</span></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Edition</span></td><td class="metadata_value"><span dir=ltr>illustrated</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>Springer Science & Business Media, 2004</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0387214283, 9780387214283</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>198 pages</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Subjects</span></td><td class="metadata_value"><div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22" itemprop="url" dir=ltr><span itemprop="title">Mathematics</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Algebra%22" itemprop="url" dir=ltr><span itemprop="title">Algebra</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Algebra+General%22" itemprop="url" dir=ltr><span itemprop="title">General</span></a></div><br><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Algebra+/+Abstract%22&source=gbs_metadata_r&cad=3"><span dir=ltr>Mathematics / Algebra / Abstract</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Algebra+/+General%22&source=gbs_metadata_r&cad=3"><span dir=ltr>Mathematics / Algebra / General</span></a><br/><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Number+Theory%22&source=gbs_metadata_r&cad=3"><span dir=ltr>Mathematics / Number Theory</span></a></td></tr><tr class="metadata_row"><td>&nbsp</td><td>&nbsp</td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Export Citation</span></td><td class="metadata_value"><a class="gb-button " href="https://books.google.com.sg/books/download/A_Field_Guide_to_Algebra.bibtex?id=Rz2VlLa3paEC&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/A_Field_Guide_to_Algebra.enw?id=Rz2VlLa3paEC&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " 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