{"id":75,"date":"2016-06-23T20:59:50","date_gmt":"2016-06-23T18:59:50","guid":{"rendered":"http:\/\/berndt-schwerdtfeger.de\/?page_id=75"},"modified":"2025-11-27T08:36:05","modified_gmt":"2025-11-27T07:36:05","slug":"algebra","status":"publish","type":"page","link":"https:\/\/berndt-schwerdtfeger.de\/?page_id=75","title":{"rendered":"Algebra"},"content":{"rendered":"<h3><a name=\"amca\"><\/a>Introduction to Commutative Algebra<br \/>\n<small>by <em>M. F. ATIYAH and I. G. MACDONALD<\/em><\/small><\/h3>\n<ul>\n<li>version 1.0, 2024-11-17 : <a title=\"amca.pdf\" href=\"wp-content\/uploads\/pdf\/amca.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>\nMSC Primary 13-01, Secondary 13A18, 13B21, 13E05, 13E10<\/li>\n<\/ul>\n<p>This digital re-issue of the book published in 1969 is meant for educational and scholarly purposes.<br \/>\nIt is typeset in TeX.<\/p>\n<h3><a name=\"c2\"><\/a>Invariants of curves of second order<\/h3>\n<ul>\n<li>version 1.0, 2013-09-13 : <a title=\"c2.pdf\" href=\"wp-content\/uploads\/pdf\/c2.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>\nMSC Primary 14H50<\/li>\n<\/ul>\n<p><em>Conics<\/em> occur as orbits of massive bodies in a gravitational field,<br \/>\nhence they are important in <em>Astronomy<\/em>.<br \/>\nThe <em>orbital elements<\/em> are fundamental invariants of the curve.<br \/>\nDetermining the invariants of curves of second order<br \/>\na<sub>11<\/sub>x<sup>2<\/sup> +2a<sub>12<\/sub>xy + a<sub>22<\/sub>y<sup>2<\/sup> + 2<sub>13<\/sub>x +2a<sub>23<\/sub>y + a<sub>33<\/sub> = 0<br \/>\nfrom the coefficients a<sub>ik<\/sub> of the equation is a well known task.<br \/>\nWe derive formulas for the invariants following an elementary high school<br \/>\ncourse developed by Freudenthal and Heinrich 50 years ago.<\/p>\n<h3>Note on adjoint functors<\/h3>\n<ul>\n<li>version 1.0, 2011-04-25 : <a title=\"adjoint.pdf\" href=\"wp-content\/uploads\/pdf\/adjoint.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>\nMSC Primary 18A40, Secondary 18A30<\/li>\n<\/ul>\n<p>Basic definition of categories, functors, morphism of functors, adjunction of functors.<\/p>\n<h3><a name=\"k2\"><\/a>Plane curves of second order and their invariants<\/h3>\n<h4>Neue Behandlung der Kurven zweiter Ordnung durch Invarianten<br \/>\n<small>von <em>Ernst Freudenthal<\/em> und <em>Werner Heinrich<\/em><\/small><\/h4>\n<ul>\n<li>Version 1.0, 2015-03-04 : <a title=\"k2.pdf\" href=\"wp-content\/uploads\/pdf\/k2.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a> (deutsch !)<br \/>\nfirst published: 2010-04-21<br \/>\nMSC Primary 14H50<\/li>\n<li>Version 1.0, 2015-03-04 : <a title=\"k2l.pdf\" href=\"wp-content\/upload\/pdf\/k2l.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a> (deutsch !) <em>L\u00f6sungsheft zu allen Aufgaben<\/em><br \/>\nfirst published: 2011-11-30<\/li>\n<\/ul>\n<p>In grateful souvenir to Werner Heinrich, who introduced me to Mathematics.<\/p>\n<p>In this course the plane curves of second order are treated by their invariants. The approach is basically algebraic, although several geometric constructions (focal generation, pencils of curves, Pascal line configuration) are given as well.<\/p>\n<p>The course contains several detailed examples and about 250 exercises. In this TeX version I have generated some 20 figures with METAPOST.<\/p>\n<p>The recent revision fixed several typos. The solution booklet to all the exercises is published as well.<\/p>\n<h3><a name=\"card\"><\/a>Cardano&#8217;sche Formeln<\/h3>\n<ul>\n<li>version 1.1, 2015-03-04 : <a title=\"card.pdf\" href=\"wp-content\/uploads\/pdf\/card.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a> (deutsch !)<br \/>\nfirst published: 2010-04-05<br \/>\nMSC Primary 12E05, Secondary 12F10<\/li>\n<\/ul>\n<p>F\u00fcr meinen Bruder <em>Klaus<\/em>.<\/p>\n<p>A short note on <em>Cardano<\/em>&#8216;s and <em>Ferrari<\/em>&#8216;s equations for solving cubic and biquadratic equations.<\/p>\n<p>Eine kurze Note \u00fcber die Formeln von <em>Cardano<\/em> und <em>Ferrari<\/em> zur L\u00f6sung von kubischen und biquadratischen Gleichungen durch Ziehen von quadratischen und kubischen Wurzeln aus den Koeffizienten.<\/p>\n<p>In der aktualisierten Version 1.1 habe ich das Vorwort korrigiert, nachdem ich die Literaturangaben \u00fcberpr\u00fcft habe (in <em>Jantzen-Schwermer<\/em> wird die <em>Cardano<\/em> Formel nur ohne Beweis angegeben, zu <em>Ferrari<\/em> steht dort nichts).<\/p>\n<h3>A note on Separability<\/h3>\n<ul>\n<li>version 1.1, 2015-03-04 : <a title=\"sep.pdf\" href=\"wp-content\/uploads\/pdf\/sep.pdf\"><img decoding=\"async\" src=\"wp-content\/uploads\/pdf.png\" alt=\"pdf\" \/><\/a><br \/>\nfirst published: 2004-01-06<br \/>\nMSC Primary 12F10, Secondary 12F15<\/li>\n<\/ul>\n<p>This note gathers key notions and proofs around separability with only modest prerequisites, covered by a first course in algebra. This revision contains only minor rearrangements vs. the original version from January 2004.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Commutative Algebra by M. F. ATIYAH and I. G. MACDONALD version 1.0, 2024-11-17 : MSC Primary 13-01, Secondary 13A18, 13B21, 13E05, 13E10 This digital re-issue of the book published in 1969 is meant for educational and scholarly purposes. It is typeset in TeX. Invariants of curves of second order version 1.0, 2013-09-13 : [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ngg_post_thumbnail":0,"footnotes":""},"class_list":["post-75","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/75","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=75"}],"version-history":[{"count":12,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/75\/revisions"}],"predecessor-version":[{"id":1045,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/75\/revisions\/1045"}],"up":[{"embeddable":true,"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/berndt-schwerdtfeger.de\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=75"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}