17 Agosto 2018, 18:14 • Fernando Abel da Conceição Silva

     <td><p><strong><font color="#2c3fb1" size="5">Welcome to Algebra</font></strong></p>     <p>The Algebra course is designed to cover the principal algebraic structures, complementing undergraduate studies in this area. Its main purpose is to prepare students to deal with Algebra that they may encounter throughout Mathematics.</p><p>As it is regulated for the Master Program  in Mathematics, if there are students who do not understand Portuguese, classes will be taught in English.</p><p>It is assumed that the contents of Algebra courses taught in undergraduate studies in Mathematics are known. In particular, Chapters 0, 1 and 2 of the <a href="http://webpages.ciencias.ulisboa.pt/~fasilva/alg3-lm/alg3.pdf">textbook</a> of <a href="http://webpages.ciencias.ulisboa.pt/~fasilva/alg3-lm/">Algebra III</a> will be used as a reference and students can use them for review. Chapters 3 and 4 of Algebra III are not needed. Other topics are also considered to be known from previous courses and students should review them as needed. The teacher will be available to assist in these reviews and, if necessary, extra classes can be scheduled for this purpose.</p><p>I recommend that students review, as soon as possible, the contents of pages 1&ndash;39 of the textbook<!--Na Faculdade de Ciências, estas matérias são lecionadas nas disciplinas de Álgebra Linear e Geometria Analítica I e II e de Álgebra I e II do primeiro ciclo.--> of Algebra III, using this or any other text. For now,  proofs can be skipped. Pages 40&ndash;80 can be reviewed later as needed.</p><p><font size="5" color="#2c3fb1">  <strong>Detailed  planned program</strong></font><strong><br></strong><!-- Due to the use of previous texts that have been adapted for this course,the first two chapters are written in English and the rest in Portuguese.--></p><p>The detailed planned program may have minor changes.</p><p><font size="4" color="#2c3fb1"><strong>1. Groups</strong></font><br>  Generalities about groups. Direct products and semidirect products. Finite <em>p</em>-groups and the <a href="https://en.wikipedia.org/wiki/Peter_Ludwig_Mejdell_Sylow">Sylow</a> theorem<a href="https://en.wikipedia.org/wiki/Peter_Ludwig_Mejdell_Sylow"></a>. Applications to the classification of finite groups. Group automorphisms. Characteristically simple groups. Groups with operators. <a href="https://en.wikipedia.org/wiki/Emmy_Noether">Noether</a>ian groups and <a href="https://en.wikipedia.org/wiki/Emil_Artin">Artin</a>ian groups. Composition series, <a href="https://en.wikipedia.org/wiki/Otto_Schreier">Schreier</a> and <a href="https://en.wikipedia.org/wiki/Camille_Jordan">Jordan</a>&ndash;<a href="https://en.wikipedia.org/wiki/Otto_H%C3%B6lder">Hölder</a> theorems. Solvable groups and nilpotent groups. <a href="https://en.wikipedia.org/wiki/Philip_Hall">Hall</a> subgroups.<br></p>
<p><font size="4" color="#2c3fb1"><strong>2. Modules</strong></font><br>Generalities about rings, modules and algebras. Chain conditions. Free modules. Exact sequences. Projective modules and injective modules. Hom functors.</p>
<p><font size="4" color="#2c3fb1"><strong>3. Multilinear maps and tensor products</strong></font><br>Multilinear maps. <a href="https://en.wikipedia.org/wiki/Arthur_Cayley">Cayley</a>&ndash;<a href="https://en.wikipedia.org/wiki/William_Rowan_Hamilton">Hamilton</a> theorem. Tensor products of modules over commutative rings. Change of the ring of scalars. Tensor products of algebras.<!-- (Produtos tensoriais de homomorfismos. Mudança do anel dos escalares. Produtos tensoriais de álgebras e anéis.) --></p><p><font size="4" color="#2c3fb1"><strong>4. Modules over principal ideal domains</strong></font><br>  Diagonalization of matrices. Classification of finitely generated modules over principal ideal domains. Classification of finitely generated <a href="https://en.wikipedia.org/wiki/Niels_Henrik_Abel">Abel</a>ian groups. Canonical forms for similarity.</p>
<p>  <font size="4" color="#2c3fb1"><strong>5. Commutative rings</strong></font><br>  Integral extensions. Prime ideals and radical ideals. <a href="https://en.wikipedia.org/wiki/Emmy_Noether">Noether</a> normalization lemma. <a href="https://en.wikipedia.org/wiki/David_Hilbert">Hilbert</a> Nullstellensatz.  Rings of fractions and localization. Primary decomposition. <a href="https://en.wikipedia.org/wiki/Emanuel_Lasker">Lasker</a>&ndash;<a href="https://en.wikipedia.org/wiki/Emmy_Noether">Noether</a> theorem.</p><p><font size="5" color="#2c3fb1"><strong>Main bibliography</strong></font></p><p><font size="4" color="#2c3fb1">&#9679;</font> Lecture notes  will be available as the classes are going on. References in the lecture notes relate to the following list.<br>  <font size="4" color="#2c3fb1">&#9679;</font> <a href="http://webpages.ciencias.ulisboa.pt/~fasilva/alg3-lm/alg3.pdf">Textbook</a> of <a href="http://webpages.ciencias.ulisboa.pt/~fasilva/alg3-lm/">Algebra III</a>. This text is being revised. Occasionally it may be corrected or clarified, but it will be avoided that the numbering of propositions is modified to keep the references clear.<br>  <font size="4" color="#2c3fb1"><strong>&#9679;</strong></font> [Brison] O. J. Brison,<em> Grupos e Representações</em>, Departamento de Matemática da FCUL, 1999.<br>  <font size="4" color="#2c3fb1"><strong>&#9679;</strong></font> [Grillet] P. Grillet, <em>Abstract Algebra</em>, Springer, 2007.<br>  <font size="4" color="#2c3fb1"><strong>&#9679;</strong></font> [Hungerford] <a href="https://en.wikipedia.org/wiki/Thomas_W._Hungerford">T. W. Hungerford</a>, <em>Algebra</em>, Springer, 1974.<br>  <font size="4" color="#2c3fb1"><strong>&#9679;</strong></font> [Lang] <a href="https://en.wikipedia.org/wiki/Serge_Lang">S. Lang</a>, <em>Algebra</em>, Springer, 2002.</p><p><font size="5" color="#2c3fb1"><strong>Evaluation</strong></font></p><p>The evaluation has two parts: a presentation and a written final exam. The presentation is optional and weighs 1/5 for the final grade if the student completes this part. The final written exam is mandatory and can be complemented by an oral examination when the teacher understands that it is necessary to confirm the mark of the written final exam.</p><p>Last update: Aug 17, 2018<br>  Fernando Silva &lt;fasilva@ciencias.ulisboa.pt&gt;</p>