Where: BUCH A 202.

When: Tue, Th 9:30-11am.

My office: Math 217.

e-mail: gor at math dot ubc dot ca

Office Hours: Tuesdays 11am-noon, Fridays 10-11am, and by appointment.

- Course outline
- The policies on marking, etc. can be found here .
- The midterm: In class on Thursday February 16 (this date is tentative and will be confirmed during the first week of class).

- Please sign up for a Piazza forum. The sign-up link . The class link.

- Here are some resources if you want to start using TeX (optional, of course).

Just for fun (optional reading, not required to solve the problem) -- Problem 7 is related to an elementary Pick's Theorem .

The last set of suggested problems -- you might want to look at them before the final exam (not to be handed in):

- Secion 10.1, proplems 3, 12, and 15-19.
- Section 12.2: 1,2,3,4,6,8,9,10,18.
- Section 12.3: 1,2,4,5,9,12,17,19,20,22.

- Tuesday Jan. 3 : The basics: motivation, and properties of the integers from a new perspective (Sections 0.2, 0.3); Rings -- the basic definitions.
- Thursday Jan. 5 : Basic properties of rings; zero divisors and units. Examples: function rings, matrix rings, Hamilton's quaternions, and quadratic integer rings. Reference: Sections 7.1 and 7.2.
- Tuesday Jan. 10 :
One more example or rings: polynomial rings (from 7.2).
Units in quadratic integer rings.

Here is a completely optional, and not to be handed in, problem set exploring solutions to Pell's equation (using continued fractions), which is equivalent to finding the units in the corresponding quadratic integer ring. - Thursday Jan. 12 : Section 7.3: homomorphisms. The notion of an ideal. Quotient rings. Examples of ring homomorphisms from 7.3.
- Tuesday Jan. 17 : The first isomorphism theorem. Examples of quotient rings: Q[x]/(x), Q[x]/(x^2+1), Q[x]/(x^2-5) (these were discussed in great detail); in particular, we discussed why the resulting rings are not isomorphic to each other (for that, we discussed a little the idea of a polnomial equation with integer coefficients having a solution in a given ring). Another example: reduction homomorphisms; relationship between having solutions in Z and solutions modulo n for all integers n. (see the discussion on p.246 in 7.3).
- Thursday Jan. 19 : The second, third, and fourth isomorphism theorems, with examples. Sums and products of ideals. We finished Section 7.3! Started 7.4 -- properties of ideals. The notion of the generating set for an ideal. Principal ideals.
- Tuesday Jan. 24 :
Finished section 7.4. Maximal ideals; prime ideals. The
criterion for being prime/maximal. Examples (in Z[x], F[x], and Z[i]).
We discussed the quotient Z[i]/(2) and concluded that the ideal (2) was
neither prime nore maximal in Z[i]. As a side note, Z[i]/(2) is a rather
peculiar ring of 4 elements.

If you are curious about how many rings of 4 elements exist, here is a general proof that there are 11 rings of order p^2, for a prime p. Later we will recognize some of these rings as quotients of polynomial rings. This is completely optional reading - Thursday Jan. 26 : Section 7.6 "Chinese remainder Theorem". did an example of a problem that amounts to Chinese Remainder Theorem for Z, with careful analysis of what the solution requires. Chinese Remainder Theorem for general rings. Started 8.1 -- defined Euclidean domains.
- Tuesday Jan. 31 :
Proof that Euclidean domains are PIDs (Principal Ideal
Domains).
Proof that Gaussian integers are a Euclidean domain, as well as the
quadratic integer ring with D=-2.
Proof that Z[\sqrt{-5}] is not a PID and
therefore, not Euclidean.
References: Section 8.1, except we skipped everything after
Theorem 4 (Please read Theorem 4 with proof; everything after it,
including Proposition 5 and the proof that
Z[1+\sqrt(-19)/2] is not Euclidean is optional), and
Section 8.2 up to (and including) Proposition 7.
We skipped everything about Dedekind-Hasse norms and the proof that
Z[1+\sqrt(-19)/2] is a PID).

Here is another reference on Euclidean rings, and this proof:

Notes on Euclidean domains by Keith Conrad (also optional reading). - Thursday Feb. 2 : Unique factorization domains - Section 8.3 up to Corollary 15 on p.289. Irreducible elements, prime elements. Discussed in detail the examples of non-unique factorizations in some quadratic integer rings, and how one gets non-principal ideals from these factorizations.
- Tuesday February 7: Finished Section 8.3 (the application to Fermat's theorem about sums of squares), and started 9.1. Please read Sections 9.1-9.2 (which are easy reading and I would like to go through them quickly). Stated Gauss' Lemma (Prop. 5 on p.303). The form stated in class was closer to Corollary 6 on p. 304 (which is in fact an equivalent reformulation).
- Thursday February 9: Started talking about polynomial rings (sketched 9.1-9.2, which were assigned as home reading). Also, discussed Section 7.5, because we need the field of fractions for the proof of Theorem 7 in 9.3.
- Tuesday February 14: Started 9.3; proved Gauss' Lemma (Proposition 5 on p.303). So far, discussed the statement and overall strategy of the proof of Theorem 7 in 9.3 (see p. 304-305). This theorem is the main result of this section, and it says that R is a UFD iff R[x] is a UFD. Will finish proving this theorem next time.
- Thursday February 16: Midterm
- Feb. 21 - 23: break.
- Tuesday Feb.28 : Finished Section 9.3, and discussed some examples -- how polynomial rings give rise to field extensions. This is covered in Section 13.1.
- Thursday March 2 :
More discussion of constructing field extensions (in particular,
constructing a finite field with a given number of elements) (see 13.1,
we will cover it in lecture up to (including) Theorem 3, but it is a
good
idea to read the whole section 13.1). Please also read 9.5.
Irreducibility criteria for polynomials (Section 9.4).

Supplementary reading if you like:

A very nice explicit description of using cyclotomic polynomials to construct finite fields appears in Section 9 here (notes by Max Naunhoffer).

A note on cyclotomic polynomials by Paul Garrett.

See also Some extra problems of number-theoretic flavour that can be solved using quadratic integer rings. - Tuesday March 7 (guest lecture) : Modules -- Sections 10.1, 10.2. The main examples: modules over a field are vector spaces; modules over Z are abelian groups. The notion of a submodule; ideals as submodules. Definition of a module homomorphism. The notions of: submodule and quotient module were defined. Also, the submodule criterion (Proposition 1 in 10.1) was proved.
- Thursday March 9: Finished the discusion of the construction of finite fields started on March 2 (see links under March 2). Continued modules -- please read all of sections 10.1, 10.2. Defined the notion of annihilator, which is discussed in Homework 8 (see the homework for the definition). Did some review of linear algebra, in particular, discussed the abstract notion of a vector space over a field (which is a special case of a module when the ring is a field).
- Tuesday March 14: Finished everything in 10.1, 10.2. In particular, returned to the discussion of modules over F[x] (namely, that a module over F[x] is a vector space with a linear operator on it) -- this is the part of 10.1 on pp.340-341. Also, defined the notions of a generating set of a module, a cyclic module, a finitely generated module. This is the first two pages of Section 10.3.
- Thursday March 16: Finished 10.2: discussed the four isomorphism theorems for modules; Defined the homomorphism group Hom(M,N) and the ring End(M). Defined the notion of an algebra and explained why End(M) is an algebra over the ring R (when M is an R-module). Also discussed direct sums and products of modules.
- Tuesday March 21:
10.3 free modules;
Examples of free and non-free modules.
The notion of rank (which is defined in 12.1).

Please read all of 10.1-10.3, with emphasis on 10.3; also please look at Problems 20, 24 and 27 in Section 10.3. - Thursday March 23: Thomas discussed the proof of Theorem 4 in 12.1; Thomas' proof was closer to the proof in the case of Euclidean rings that is outlined in Problems 16-19 in 12.1. -- please look at these exercises and read the proof of Theorem 4, and Theorems 5-7.
- Tuesday March 28:
Will finish the discussion of free modules from last Tuesday, especially
the universal property, and will finish the discussion of when the notion
of rank is (and isn't) well-defined. -- finish Section 10.3.
Then we get back to discussing the classification of modules over PIDs
(Section 12.1)

For a lot more information on universal properties, see Supplement on categories by Lior Silberman.