MATD01 Fields and Groups

Winter 2020

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.

Course Description

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

Teaching Staff

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

Course Materials

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

Midterm 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

Weekly Calendar

Week 1

Reading: Review of your group theory course, Ch. 2 and 3 of Rotman

What we did:

• Monday: We introduced the notion of a ring and saw some examples.
• Wednesday: We introduced the notions of a subring, zero divisor, integral domain, unit and field. We saw that the set of zero divisors and units in a ring are disjoint (and in particular, being a unit, any nonzero element of a field is not a zero divisor, so that any field is an integral domain). We saw that the set of units in a ring R form a group under multiplication. We calculated this group (called the group of units) for Z/nZ, and saw that Z/nZ is a field if and only if n is a prime number. Finally we defined the polynomial ring R[x] for any given ring R.

Next week's plan: Chapters 4 and 5

Week 2

Reading: Ch. 4 and 5 of Rotman

What we did:

• Monday: We introduced ring homomorphisms and ideals. We saw that the kernel of a ring homomorphism is an ideal (of the domain). Also that a ring homomorphism is injective if and only if its kernel is zero.
• Wednesday: In the first part of the lecture we continued our discussion of ideals. We saw the ideals of Z are of the form n Z. We introduced the notion of the ideal generated by an element. We showed that a ring R is a field if and only if the only ideals of R are zero and R. We showed that any ring homomorphism whose domain is a field is injective. In the second part of the lecture we defined quotient rings.

Next week's plan: Ch. 5 (continued), Ch. 6

Week 3

Reading: Ch. 5 and Ch. 6 of Rotman

What we did:

• Monday: We talked about the first isomorphism theorem. Then we started Ch. 6 and proved the division algorithm.
• Wednesday: We discussed evaluation maps R[x]→ R for a ring R. We used the division algorithm to show that an element α of R is a root of f(x)∈R[x] if and only if x-α divides f(x). Also that if R is a domain and f(x)∈R[x] has distinct roots α₁,...,α_n then the product (x-α₁)...(x-α_n) divides f(x). As a corollary we concluded that if R is a domain, a polynomial of degree n in R[x] can have at most n distinct roots in R. We then showed that is F is a field, the ring F[x] is a PID (that is, an integral domain in which every ideal is principle). We defined the gcd of two polynomials in F[x] and stated the result that given polynomials f(x),g(x)in F[x] with at least one nonzero, their gcd exists and is unique. We proved the uniqueness. We'll do existence at the beginning of the next lecture.

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.)

Week 4

Reading: Ch. 6 of Rotman

What we did:

• Monday: We finished the proof of existence and uniqueness of gcd in F[x]. In the process of the proof we saw that the gcd of f and g is the monic generator of the ideal (f,g). We then recalled Euclid's algorithm for finding the gcd. We noted that the calculations of the algorithm can be used to write the gcd of two polynomials as a linear combination of them.

• 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)^m_i, 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

Week 5

Reading: Ch. 7 of Rotman

What we did:

• Monday: We talked about prime and maximal ideals. The main results were: (1) An ideal I of a ring R is prime (resp. maximal) if and only if R/I is an integral domain (resp. a field). (2) Any maximal ideal is prime. (3) If a principal ideal (r) in an integral domain is prime, then r is irreducible. (4) In a PID, the following are equilavent for any ideal I: (i) I is nonzero and prime; (ii) I is nonzero and maximal; (iii) I=(r) for an irreducible r.

• 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 α₁,...,α_n be the distinct roots of f(x) in K. Then F(α₁,...,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

Week 6

Reading: Ch. 7 and Ch. 8 of Rotman

What we did:

• Monday: We finished the proof of existence of finite fields of any prime power order. We discussed the notion of repeated roots. We defined the derivative of a polynomial and used it to give a criterion for a polynomial to have repeated roots. As an example we talked about when the polynomial xⁿ-1∈F[x] (n>0) has repeated roots. (If the characteristic of F does not divide n (in particular, when F is of characteristic 0), then xⁿ-1 does not have repeated roots. If char(F)=p > 0 divides n, then xⁿ-1=(x^n/p-1)^p and hence always any of its roots is a repeated root. Writing n=p^an' with n' not a multiple of p, we have xⁿ-1=(x^n'-1)^pa. Thus the roots of xⁿ-1 in any field extension K/F are exactly the roots of x^n'-1. The latter polynomial has no repeated roots (because p does not divide n'). If x^n'-1 splits over K with roots α₁,...,α_n' (which are all distinct), then xⁿ-1 factors as xⁿ-1 = ( ∏ ^n' _i=1 (x-α_i) )^pa .)

• 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.

Week 8

Reading: Ch. 9 and 10 of Rotman

What we did:

• Monday: We discussed the quadratic and cubic formulas.

• 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

Week 9

Reading: Ch. 10 of Rotman

What we did:

• Monday: We did another couple of examples for the degree formula. We saw as applications that the field obtained from F by adjoining finitely many algebraic elements is a finite extension of F.

• 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

Week 10

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.

Week 11

Reading: Ch. 12 of Rotman

What we did: See Bb Collaborate for the recordings of the lectures.

Week 12 (last week of classes)

Reading: Ch. 13 of Rotman

What we did: See Bb Collaborate for the recordings of the lectures.