{"id":125,"date":"2017-01-10T12:46:07","date_gmt":"2017-01-10T12:46:07","guid":{"rendered":"https:\/\/sites.wcsu.edu\/roccac\/?page_id=125"},"modified":"2017-05-24T15:01:40","modified_gmt":"2017-05-24T15:01:40","slug":"modern-algebra-ii","status":"publish","type":"page","link":"https:\/\/sites.wcsu.edu\/roccac\/homepage\/modern-algebra-ii\/","title":{"rendered":"Modern Algebra II"},"content":{"rendered":"<ul>\n<li><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2017\/01\/2017-Spring-MAT-513-Syl.pdf\" target=\"_blank\">Syllabus<\/a><\/li>\n<li>Class Calendar:\n<ol>\n<li>Canceled Due to Weather (1\/23\/2017)<\/li>\n<li>Syllabus, introduction, historical notes, and Cardano&#8217;s Formula (1\/30\/2017)<\/li>\n<li>Covered material from chapters 3 and 4 (2\/6\/2017)<\/li>\n<li>Covered some number theory and basics from chapter 5 (2\/13\/2017)<\/li>\n<li>Spent time discussing field extensions, their degree, bases for them, and whether or not they are simple. (Chapter 6) (2\/27\/2017)<\/li>\n<li>We covered the highlights from 8.1 through 8.6 and looked at a key lemma from section 8.7.\u00a0 Monday after break is your first exam and then we move forward the following week. (3\/6\/2017)<\/li>\n<li>Exam 1: Preliminaries and Field Extensions (3\/20\/2017)<\/li>\n<li>Finished Chapter 8 (3\/27\/2017), Read 8.3 pp. 110-111 in particular<\/li>\n<li>Did an overview of chapters 9 through 11 (4\/3\/2017)<\/li>\n<li>Went over the example in chapter 13 (4\/10\/2017)<\/li>\n<li>plan to look at 12-14 (4\/17\/2017)<\/li>\n<li>exam on 8-11 (4\/24\/2017)<\/li>\n<li>plan to look at 14-15 (5\/1\/2017)<\/li>\n<li>exam 12-15 (5\/8\/2017)<\/li>\n<\/ol>\n<\/li>\n<li>Assignments: For each chapter complete the true\/false question and two others from the following:\n<ul>\n<li>Chapter 1: 1.3, 1.7, 1.9, 1.13<\/li>\n<li>Chapter 2: 2.1, 2.2, 2.4<\/li>\n<li>Chapter 3: 3.2-3.3 (as a single problem), 3.5-3.6 (as one), 3.7<\/li>\n<li>Chapter 4: 4.1, 4.3(c,e,g), 4.4, 4.7 (inverse means reciprocal)<\/li>\n<li>Chapter 5: 5.1, 5.3, 5.4, 5.6, 5.8<\/li>\n<li>Chapter 6:6.1(b,d,f), 6.3 (you may want to think about 6.4 to help with this), 6.5, 6.9, 6.12<\/li>\n<li>Chapter 8: 8.1, 8.2, 8.3, 8.4, 8.7, (and remember to do the true false)<\/li>\n<li>Chapter 9: 9.1 &amp; 9.2 (together), 9.5, 9.6, (and remember to do the true false)<\/li>\n<li>Chapter 10: 10.2, 10.3, (and remember to do the true false)<\/li>\n<li>Chapter 11: 11.1, 11.2-11.4 parts (a) and (d), 11.6, (and remember to do the true false)<\/li>\n<\/ul>\n<\/li>\n<li>Assignments for Chapters 12-15: Do one problem from each chapter, you may do two additional problems for extra credit. Turn it in by the end of final exam week.\n<ul>\n<li>Chapter 12: 12.1, 12.2 (Hint: &#8230;), 12.7 (assuming you are given 12.6)<\/li>\n<li>Chapter 13: 13.1-13.5 (c), 13.9<\/li>\n<li>Chapter 14: 14.1 (work from the definition of solvable), 14.4, 14.10<\/li>\n<li>Chapter 15: 15.1, 15.2, 15.3 (just do one of the polynomials), 15.7 (look at lemma 15.3 and theorem 14.4)<\/li>\n<\/ul>\n<\/li>\n<li>Exams\/Quizzes:\n<ul>\n<li>History &amp; Polynomials &amp; Field Extensions: 3\/20\/2017\n<ul>\n<li>Definitions: Cardano&#8217;s Formula, Irreducible Polynomial, Minimal Polynomial, Field, Field Extension, Simple Extension, Algebraic Extension, Trancendental Extension, Degree of an Extension, Vector Space, Basis for a Vector Space.<\/li>\n<li>Theorems: Be sure that you know these and you should be prepared to prove some of them\n<ul>\n<li>Eisenstein&#8217;s Criterion<\/li>\n<li>Minimal Poly iff Irreducible Polynomial.<\/li>\n<li>Minimal Polynomial divides all Polynomials with the same root.<\/li>\n<li>Every element of \\(K[t]\/&lt;m&gt;\\) has an inverse iff \\(m\\) is an irreducible polynomial.<\/li>\n<li>If \\(\\alpha\\) has minimal polynomial \\(m\\), then \\[(K[t]\/&lt;m&gt;):K \\cong K[\\alpha]:K.\\]<\/li>\n<li>Let \\(K(\\alpha):K\\) be a simple algebraic extention, let the minimal polynomial of \\(\\alpha\\) over \\(K\\) be \\(m\\), and let \\(\\partial m = n\\) (the degreee of \\(m\\)).\u00a0 Then \\(\\{1,\\alpha,\\alpha^2,\\ldots,\\alpha^{n-1}\\}\\) is a basis for \\(K(\\alpha)\\) over \\(K\\).<\/li>\n<li>Tower Theorem: If \\(K,L,M\\) are subfields of \\(\\mathbb{C}\\) with \\(K\\subseteq L\\subseteq M\\), then \\[[M:K]=[M:L]\\cdot[L:K].\\]<\/li>\n<\/ul>\n<\/li>\n<li>Skills:\n<ul>\n<li>Depress a Cubic.<\/li>\n<li>Use Cardano&#8217;s formula to factor a cubic.<\/li>\n<li>Determine\/Show that a polynomial is irreducible.<\/li>\n<li>Determine the degree of a field extension<\/li>\n<li>Find a basis for a field extension<\/li>\n<\/ul>\n<\/li>\n<li>Practice Problems:\n<ul>\n<li>Look at all the true false questions.<\/li>\n<li>Chap. 1: 1.2,1.5,1.7;<\/li>\n<li>Chap. 2: 2.4;<\/li>\n<li>Chap. 3: 3.1-3.5,3.8;<\/li>\n<li>Chap. 4: 4.1-4.4;<\/li>\n<li>Chap. 5: 5.1-5.4, 5.6, 5.7;<\/li>\n<li>Chap. 6: 6.1-6.3, 6.7(use the contrapositive and the Tower Theorem), 6.9, 6.12(look at 6.3 and use proof by contradiction), 6.13.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Basics of Galois Theory, <span style=\"text-decoration: underline\"><em>Chapters 8-11 Exam on 4\/24\/2017<\/em><\/span>\n<ul>\n<li>Definitions: Irreducible Polynomial, Minimal Polynomial, Field, Field Extension, Simple Extension, Algebraic Extension, Degree of an Extension, Vector Space, Basis for a Vector Space, Galois Group, \\(\\mathbb{S}_n\\), \\(\\mathbb{A}_n\\), Elementary Symmetric Polynomials, General Polynomial, Fixed Field, Splitting Field, Normal Extension, Normal Closure, Separable, &#8230;<\/li>\n<li>Theorems:\n<ul>\n<li>Theorem 8.10 &#8211; Insolubility of Quintics by Ruffini Radicals (know it, not prove it)<\/li>\n<li>Theorem 9.4 &#8211; The uniqueness of splitting fields for a given polynomial \\(f\\) over a subfield \\(K\\) of \\(\\mathbb{C}\\).<\/li>\n<li>Theorem 9.9 &#8211; A field extension \\([L:K]\\) is normal and finite if and only if \\(L\\) is the splitting field for some polynomial over \\(K\\).<\/li>\n<li>Theorem 10.5 &#8211;\u00a0Let \\(G\\) be a finite subgroup of the group of automorphisms of a field \\(K\\), and let \\(K_0\\)\u00a0be the fixed field of \\(G\\). Then \\[[K:K_0]=|G|.\\]<\/li>\n<li>Corollary 11.11 &#8211; If \\(L:K\\) is a finite normal extension inside \\(\\mathbb{C}\\), then there are precisely \u00a0\\([L:K]\\) distinct \\(K\\)-automorphisms of \\(L\\). \u00a0That is \\[|\\Gamma(L:K)|=[L:K]\\]<\/li>\n<\/ul>\n<\/li>\n<li>Skills:\n<ul>\n<li>Describe elements of a particular Galois Group (see examples 8.4 p.113)<\/li>\n<li>Find splitting fields (see examples 9.2,p.129 and 9.7, p.132)<\/li>\n<li>Example 10.7 p.142, relating orders of groups to degrees of externsions<\/li>\n<li>Find normal closures (see example 11.7, p.147)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Fundamental Theorem of Galois Theory And Insoluble Quintic: <span style=\"text-decoration: underline\">Chapter 12 through 15, 5\/8\/2017 @ 5:20<\/span>\n<ul>\n<li>Definitions:\u00a0Irreducible Polynomial, Minimal Polynomial, Degree of an Extension, Galois Group, \\(\\mathbb{S}_n\\), \\(\\mathbb{A}_n\\), Fixed Field, Splitting Field, Normal Extension, Normal Closure, Separable, Simple group, Solvable (soluble) group, Dihedral group of order n \\(\\left(D_n\\right)\\), \\(K\\)-automorphisms, Normal subgroup, Radical extension, Solvable by radicals, &#8230;<\/li>\n<li>Theorems: Mostly just be prepared to state and apply these theorems, some have special instructions (see below).\n<ul>\n<li>Cauchy&#8217;s Theorem (Thm 14.15, p.167)<\/li>\n<li>Theorem 12.2 Fundamental Theorem of Galois Theory, p. 151<\/li>\n<li>Theorem 14.4, p.162<\/li>\n<li>Theorem 14.6, 14.7 and Corollary 14.8, p.164-166 (In particular be sure you know how 14.4, 14.6, and 14.7 fit together to give us 14.8)<\/li>\n<li>Theorem 15.2, p.173<\/li>\n<li>Lemma 15.6, p.174 (you should be able to outline the proof of this as we did in class)<\/li>\n<li>Lemma 15.9, p.176\u00a0(you should be able to outline the proof of this as we did in class)<\/li>\n<li>Theorem 15.10, p.177 (you should be able to outline the proof of this as we did in class)<\/li>\n<\/ul>\n<\/li>\n<li>Skills:\n<ul>\n<li>Given a polynomial and its roots be able to find the splitting field.<\/li>\n<li>Given a polynomial and its roots be able to find the Galois group.<\/li>\n<li>Find the subgroups of a given Galois group.<\/li>\n<li>Given a subgroup of the Galois group be able to find the fixed field corresponding to the subgroup.<\/li>\n<li>Determine if a field extension is normal and if not find the normal closure.<\/li>\n<li>Given a quintic which is not solvable, prove it.<\/li>\n<li>Look through the homework exercise for chapters 12 through 15, particularly the more algorithmic ones.<\/li>\n<li>Be prepared to answer the true\/false questions from chapters 12-15.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Links and Handouts:\n<ul>\n<li><a href=\"http:\/\/groupexplorer.sourceforge.net\/\" target=\"_blank\">Group Explorer<\/a><\/li>\n<li><a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2016\/08\/Typesetting.pdf\" target=\"_blank\">Typesetting Mathematics<\/a><\/li>\n<li>Problem 1.3 <a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2017\/03\/Stewart-Problem-1-3.docx\" target=\"_blank\">Word Document<\/a><\/li>\n<li>Problem 1.9 <a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2017\/03\/problem-1-9.pdf\" target=\"_blank\">PDF Document<\/a> (<a href=\"https:\/\/sites.wcsu.edu\/roccac\/wp-content\/uploads\/sites\/33\/2017\/03\/problem-1-9-Source.pdf\" target=\"_blank\">\\(\\LaTeX\\) source<\/a>)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Syllabus Class Calendar: Canceled Due to Weather (1\/23\/2017) Syllabus, introduction, historical notes, and Cardano&#8217;s Formula (1\/30\/2017) Covered material from chapters 3 and 4 (2\/6\/2017) Covered some number theory and basics from chapter 5 (2\/13\/2017) Spent time discussing field extensions, their &hellip; <a href=\"https:\/\/sites.wcsu.edu\/roccac\/homepage\/modern-algebra-ii\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":35,"featured_media":0,"parent":2,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"wpo365_audiences":[],"wpo365_private":false,"footnotes":""},"class_list":["post-125","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages\/125","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/users\/35"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/comments?post=125"}],"version-history":[{"count":0,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages\/125\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/pages\/2"}],"wp:attachment":[{"href":"https:\/\/sites.wcsu.edu\/roccac\/wp-json\/wp\/v2\/media?parent=125"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}