• 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.