- Syllabus
- Class Calendar:
- Canceled Due to Weather (1/23/2017)
- Syllabus, introduction, historical notes, and Cardano’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 degree, bases for them, and whether or not they are simple. (Chapter 6) (2/27/2017)
- 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)
- Exam 1: Preliminaries and Field Extensions (3/20/2017)
- Finished Chapter 8 (3/27/2017), Read 8.3 pp. 110-111 in particular
- Did an overview of chapters 9 through 11 (4/3/2017)
- Went over the example in chapter 13 (4/10/2017)
- plan to look at 12-14 (4/17/2017)
- exam on 8-11 (4/24/2017)
- plan to look at 14-15 (5/1/2017)
- 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.

- History & Polynomials & Field Extensions: 3/20/2017
- Links and Handouts:
- Group Explorer
- Typesetting Mathematics
- Problem 1.3 Word Document
- Problem 1.9 PDF Document (\(\LaTeX\) source)