Modern Algebra II

  • Syllabus
  • Class Calendar:
    1. Canceled Due to Weather (1/23/2017)
    2. Syllabus, introduction, historical notes, and Cardano’s Formula (1/30/2017)
    3. Covered material from chapters 3 and 4 (2/6/2017)
    4. Covered some number theory and basics from chapter 5 (2/13/2017)
    5. Spent time discussing field extensions, their degree, bases for them, and whether or not they are simple. (Chapter 6) (2/27/2017)
    6. We covered the highlights from 8.1 through 8.6 and looked at a key lemma from section 8.7.  Monday after break is your first exam and then we move forward the following week. (3/6/2017)
    7. Exam 1: Preliminaries and Field Extensions (3/20/2017)
    8. Finished Chapter 8 (3/27/2017), Read 8.3 pp. 110-111 in particular
    9. Did an overview of chapters 9 through 11 (4/3/2017)
    10. Went over the example in chapter 13 (4/10/2017)
    11. plan to look at 12-14 (4/17/2017)
    12. exam on 8-11 (4/24/2017)
    13. plan to look at 14-15 (5/1/2017)
    14. exam 12-15 (5/8/2017)
  • Assignments: For each chapter complete the true/false question and two others from the following:
    • Chapter 1: 1.3, 1.7, 1.9, 1.13
    • Chapter 2: 2.1, 2.2, 2.4
    • Chapter 3: 3.2-3.3 (as a single problem), 3.5-3.6 (as one), 3.7
    • Chapter 4: 4.1, 4.3(c,e,g), 4.4, 4.7 (inverse means reciprocal)
    • Chapter 5: 5.1, 5.3, 5.4, 5.6, 5.8
    • 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
    • Chapter 8: 8.1, 8.2, 8.3, 8.4, 8.7, (and remember to do the true false)
    • Chapter 9: 9.1 & 9.2 (together), 9.5, 9.6, (and remember to do the true false)
    • Chapter 10: 10.2, 10.3, (and remember to do the true false)
    • Chapter 11: 11.1, 11.2-11.4 parts (a) and (d), 11.6, (and remember to do the true false)
  • 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.
    • Chapter 12: 12.1, 12.2 (Hint: …), 12.7 (assuming you are given 12.6)
    • Chapter 13: 13.1-13.5 (c), 13.9
    • Chapter 14: 14.1 (work from the definition of solvable), 14.4, 14.10
    • 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)
  • Exams/Quizzes:
    • History & Polynomials & Field Extensions: 3/20/2017
      • Definitions: Cardano’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.
      • Theorems: Be sure that you know these and you should be prepared to prove some of them
        • Eisenstein’s Criterion
        • Minimal Poly iff Irreducible Polynomial.
        • Minimal Polynomial divides all Polynomials with the same root.
        • Every element of \(K[t]/<m>\) has an inverse iff \(m\) is an irreducible polynomial.
        • If \(\alpha\) has minimal polynomial \(m\), then \[(K[t]/<m>):K \cong K[\alpha]:K.\]
        • 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 \(m\)).  Then \(\{1,\alpha,\alpha^2,\ldots,\alpha^{n-1}\}\) is a basis for \(K(\alpha)\) over \(K\).
        • Tower Theorem: If \(K,L,M\) are subfields of \(\mathbb{C}\) with \(K\subseteq L\subseteq M\), then \[[M:K]=[M:L]\cdot[L:K].\]
      • Skills:
        • Depress a Cubic.
        • Use Cardano’s formula to factor a cubic.
        • Determine/Show that a polynomial is irreducible.
        • Determine the degree of a field extension
        • Find a basis for a field extension
      • Practice Problems:
        • Look at all the true false questions.
        • Chap. 1: 1.2,1.5,1.7;
        • Chap. 2: 2.4;
        • Chap. 3: 3.1-3.5,3.8;
        • Chap. 4: 4.1-4.4;
        • Chap. 5: 5.1-5.4, 5.6, 5.7;
        • 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.
    • Basics of Galois Theory, Chapters 8-11 Exam on 4/24/2017
      • 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, …
      • Theorems:
        • Theorem 8.10 – Insolubility of Quintics by Ruffini Radicals (know it, not prove it)
        • Theorem 9.4 – The uniqueness of splitting fields for a given polynomial \(f\) over a subfield \(K\) of \(\mathbb{C}\).
        • Theorem 9.9 – A field extension \([L:K]\) is normal and finite if and only if \(L\) is the splitting field for some polynomial over \(K\).
        • Theorem 10.5 – Let \(G\) be a finite subgroup of the group of automorphisms of a field \(K\), and let \(K_0\) be the fixed field of \(G\). Then \[[K:K_0]=|G|.\]
        • Corollary 11.11 – If \(L:K\) is a finite normal extension inside \(\mathbb{C}\), then there are precisely  \([L:K]\) distinct \(K\)-automorphisms of \(L\).  That is \[|\Gamma(L:K)|=[L:K]\]
      • Skills:
        • Describe elements of a particular Galois Group (see examples 8.4 p.113)
        • Find splitting fields (see examples 9.2,p.129 and 9.7, p.132)
        • Example 10.7 p.142, relating orders of groups to degrees of externsions
        • Find normal closures (see example 11.7, p.147)
    • Fundamental Theorem of Galois Theory And Insoluble Quintic: Chapter 12 through 15, 5/8/2017 @ 5:20
      • Definitions: Irreducible 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, …
      • Theorems: Mostly just be prepared to state and apply these theorems, some have special instructions (see below).
        • Cauchy’s Theorem (Thm 14.15, p.167)
        • Theorem 12.2 Fundamental Theorem of Galois Theory, p. 151
        • Theorem 14.4, p.162
        • 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)
        • Theorem 15.2, p.173
        • Lemma 15.6, p.174 (you should be able to outline the proof of this as we did in class)
        • Lemma 15.9, p.176 (you should be able to outline the proof of this as we did in class)
        • Theorem 15.10, p.177 (you should be able to outline the proof of this as we did in class)
      • Skills:
        • Given a polynomial and its roots be able to find the splitting field.
        • Given a polynomial and its roots be able to find the Galois group.
        • Find the subgroups of a given Galois group.
        • Given a subgroup of the Galois group be able to find the fixed field corresponding to the subgroup.
        • Determine if a field extension is normal and if not find the normal closure.
        • Given a quintic which is not solvable, prove it.
        • Look through the homework exercise for chapters 12 through 15, particularly the more algorithmic ones.
        • Be prepared to answer the true/false questions from chapters 12-15.
  • Links and Handouts: