This is the webpage for MATD01 during Winter 2020 (UTSC campus). All the course documents will be posted here. We will be using Quercus for the purposes of announcements and recording grades.
The ultimate goal of the course is to prove that there is no general formula for the roots of a polynomial of degree ≥ 5. The theory used to prove this statement is called Galois theory (named after the French mathematician Évariste Galois (1811-1832)). Before we get to Galois theory, we need to learn about rings and fields. We will also learn some more about groups (note that we will assume that students are comfortable with group theory to the extent covered by MATC01).
Syllabus (Updated after a new grading scheme was approved by the majority of the class in a vote that took place between March 21-24 on Quercus.)
Textbook: Galois Theory by Joseph Rotman, Second Edition
Instructor: Payman Eskandari , email: payman@math.utoronto.ca
Office hours: Mondays 1:30-3:30 in IC412
TA: Pourya Memarpanahi, email: pourya.memarpanahi@utoronto.ca
Note that in addition to the mandatory part of the assignments which are to be handed in for grading, the pdfs include a list of extra practice problems as well. All assignments are to be submitted on Crowdmark.
Assignment 1 due Fri Jan 17, 10:00 pm Solutions
Assignment 2 due Fri Jan 24, 10:00 pm Solutions
Assignment 3 due Fri Jan 31, 10:00 pm Solutions
Assignment 4 due Fri Feb 7, 10:00 pm Hint for Problem 7 , Solutions
Assignment 5 due Wed Feb 19, 10:00 pm Solutions
Assignment 6 due Sat Mar 7, 10:00 pm Solutions
Assignment 7 due Sun Mar 15, 10:00 pm (File updated on Monday Mar 9. Changes: Typo in Question 1 corrected. Also a hint added for the same question.) Solutions
Assignment 8 due Sun Mar 22, 10:00 pm Solutions
Assignment 9 due Sun Mar 29, 10:00 pm Solutions
Takehome Final Exam due Wed Apr 15 at noon
Reading: Review of your group theory course, Ch. 2 and 3 of Rotman
What we did:
Next week's plan: Chapters 4 and 5
Reading: Ch. 4 and 5 of Rotman
What we did:
Next week's plan: Ch. 5 (continued), Ch. 6
Reading: Ch. 5 and Ch. 6 of Rotman
What we did:
Next week's plan: finishing Ch. 6 on Monday, Ch. 7 on Wednesday (Some parts of Ch. 7 have already been covered on the assignments.)
Reading: Ch. 6 of Rotman
What we did:
Wednesday: Up to this point we had only discussed gcd for a polynomial ring F[x]. We generalized the notion to arbitrary PIDs, defining the gcd of elements r and s of a PID as a generator of the ideal (r,s) generated by r and s. The gcd of r and s is unique up to scaling by units. (In the situation of F[x] one can (and following Rotman we earlier did) additionally require the gcd to be monic, so that it really becomes unique.) We said r and s are called relatively prime if (r,s)=R (= (1) ). We proved the following proposition: Let R be a PID and r,s,t ∈ R. Then: (i) if r and s are relatively prime and r divides st, then r divides t; (ii) if r is irreducible, then given any s either r divides s or r and s are relatively prime; (iii) if r is irreducible and it divides st, then r divides s or t.
We then proved the unique factorization theorem for polynomial rings: Let F be a field. Any nonzero f(x) in F[x] can be expressed as a product u∏_{i=1}^{k} g_{i}(x)^{mi}, where u is a unit, the g_{i}(x) are distinct monic irreducible polynomials, and the exponents m_{i} are positive integers (note that k is allowed to be zero, in which case the empty product is by definition equal to 1). Moreover, this factorization is unique up to permuting the g_{i}(x). (Existence was proved by induction on the degree of f(x), and uniqueness was an application of statement (iii) of the proposition above.
We spent the last bit of the lecture going over one of the homework questions (regarding homomorphisms from Z[x] and Q[x]).
Next week's plan: Ch. 7, starting Ch. 8
Reading: Ch. 7 of Rotman
What we did:
Wednesday: We proved that given any irreducible f(x) in F[x], there is a field extension K/F in which f(x) has a root. (Summary of proof: Take K to be the field F[x]/(f(x)). Then x ( = the image of x in K) satisfies f(x)=0.) Combining this and factorization into a product of irreducibles, we showed by induction on the degree that for any f(x) in F[x], there is a field extension K/F over which f(x) splits (i.e. all its irreducible factors are of degree 1). We then defined the notion of a splitting field, and proved that it always exists. (Summary of proof: Given f(x) in F[x], let K be a field extension of F where f(x) splits. Let α_{1},...,α_{n} be the distinct roots of f(x) in K. Then F(α_{1},...,α_{n}) (which by definition is the intersection of all subfields of K which contain F and all the α_{i}) is a splitting field of f(x) over F.) As an application of all of this, we (almost) proved that given any prime power q, there is a field with q elements. (First thing next Monday we will finish this proof.)
Next week's plan: finishing Ch. 7, Ch. 8
Reading: Ch. 7 and Ch. 8 of Rotman
What we did:
Wednesday: Chapter 8.
Next week's plan: The week of Feb 17 is the reading week and there are no classes. During the week of Feb 24 on Monday we will likely do a review of certain things so far (especially material that was introduced on the assignments). On Wednesday Feb 26 we will have our midterm in class.
Reading: Ch. 9 and 10 of Rotman
What we did:
Wednesday: We discussed the quartic formula. Then we started Ch. 10. Some parts of this chapter have been already covered in the course, such the notions of finite extensions, degree of an extension, algebraic elements, minimal polynomials, and that if α is algebraic over F then the degree of F(α)/F equals the degree of the minimal polynomial of α over F. We introduced the notion of an algebraic extension and showed that any finite extension is algebraic. We then proved the main result of the lecture: if K/F and L/K are finite extensions, then L/F is finite and [L:F]=[L:K][K:F] (we call this the degree formula). We then did two examples.
Next week's plan: finishing Ch. 10, some of Ch. 11
Reading: Ch. 10 of Rotman
What we did:
Wednesday: We showed that if f∈F[x] has degree n and K is a splitting field of f over F, then the degree [K:F] divides n!. We saw that if K/F and K'/F' are field extensions, σ:F → F' is a homomorphism, and σ^: K→K' is a homomorphism extending σ, then for any root α∈ K of a polynomial f∈ F[x], σ^(α) is a root of σ*(f) ( = the polynomial obtained by applying σ to the coefficients of f). On the other hand, if f is irreducible, for any root α' of σ*(f), there is a unique homomorphism σ^: F(α)→K' extending σ which sends α to α'.
Next week's plan: finishing Ch. 10, Ch. 11
Reading: Ch. 10, 11 of Rotman
What we did: No physical classes from now until the end of the term (due to University closure). We posted pre-recorded lectures online (see the Quercus announcement for links). We finished Ch. 10 and covered Ch. 11. We also spent some time on the Galois group of the splitting field of a cubic.
Next week's plan: We will stop posting pre-recorded lectures and instead live-stream lectures on Bb Collaborate (during the usual class times). The main plan is to cover Ch. 12.Reading: Ch. 12 of Rotman
What we did: See Bb Collaborate for the recordings of the lectures.
Next week's plan: Ch. 13 of RotmanReading: Ch. 13 of Rotman
What we did: See Bb Collaborate for the recordings of the lectures.