Math 200b Winter 2015

Note that the last day of the quarter, Friday 3/13, class will not be held.

Instructors

Professor:

Teaching Assistant:

Course Description and Syllabus:

Math 200b Winter 2015 course syllabus

Lecture Summaries

Below, we will post short summaries of what was covered in each lecture.


• Lecture 1 (1/5/15): (Section 10.1-10.2) Definition of a (left) module over a ring R. Examples. Modules over a field K are vector spaces and modules over Z are Abelian groups. Submodules, homomorphisms, factor modules, isomorphism theorems.
• Lecture 2 (1/7/15): (Section 10.2-10.3) The set Hom_R(M,N) of all R-module homomorphisms from M and N is always an Abelian group. If R is commutative it is also a left R-module. End_R(M) = Hom_R(M,M) is always a ring with function composition for the multiplication. End(R) is isomorphic to R if R is commutative. There is a one-to-one correspondence between R-module structures on an abelian group M and ring homomorphisms from R to End_Z(M). If F is a field, an F[x]-module is the same as an F-vector space V together with a linear transformation phi in End_F(V) such that x acts as phi.
• Lecture 3 (1/9/15): (Section 10.3) Direct sums and direct products of modules. The submodule generated by a subset. Free modules and their universal property.
• Lecture 4 (1/12/15): (Section 12.1) Statement of the fundamental theorem of finitely genrated modules over PIDs. Outline of proof, assuming the following technical lemma: If M is a f.g. free module over a PID R, and N is a submodule of M, then there is a basis m_1, ..., m_n of M and nonzero a_i in R such that a_1m_1, ... a_d m_d is a basis of N for some d less than or equal to n. In particular, N is free.
• (1/14/15): (No class: Professor Rogalski out of town)
• (1/16/15): (No class: Professor Rogalski out of town)
• (1/19/15): (No class: MLK Day)
• Lecture 5 (1/21/15): (Section 12.1-12.2) Sketch of the uniqueness part of the fundamental theorem of finitely generated modules over PIDs. Review of how matrices correspond to linear transformations. Beginning of discussion of rational canonical form of a linear transformation.
• Lecture 6 (1/23/15): (Section 12.2) More on the rational canonical form. The minimal and characteristic polynomial and their relation to the rational form and the invariant factors. The Cayley-Hamilon Theorem. Similarity of matrices is independent of the field.
• Lecture 7 (1/26/15): (Section 12.3) Jordan canonical form over an algebraically closed field. Relation to generalized eigenspaces.
• Lecture 8 (1/28/15): (Section 10.4) Tensor products. The universal property of a tensor product. Existence of the tensor product. Module structure of M \otimes_R N in case M or N is a bimodule.
• Lecture 9 (1/30/15): (Section 10.4) More on tensor products. Given a homomorphism of rings R to S, we can extend any left R-module M to get a left S-module S \otimes_R M. If R is commutative and M and N are R-modules, M \otimes_R N is universal for R-bilinear maps. Direct sums pull out of either coordinate of a tensor product.
• Lecture 10 (2/2/15): (Section 10.4-10.5) R-algebras. The tensor product of two R-algebras is an R-algebras. Example: K[x] \otimes_K K[y] is isomorphic to K[x,y]. Introduction to homological algebra. Short exact sequences. Split short exact sequences.
• Lecture 11 (2/4/15): (Section 10.5) Right exactness of tensor product. Left exactness of Hom. Projective modules.
• Lecture 12 (2/6/15): (Section 10.5) Projective modules are the same as direct summands of free modules. Examples of projective modules. Injective modules. E is injective if and only if Hom(-, E) is exact, and P is projective if and only if Hom(P, -) is exact. Flat modules. Projective modules are flat.
• Lecture 13 (2/9/15): (Section 10.5) More on injective modules. The Baer criterion. Injective modules are divisible. Divisible modules over a PID are injective. Examples of injective modules. The adjointness of Hom and tensor.
• Lecture 14 (2/11/15): (Section 13.1) Introduction to field theory. Two ways to produce new fields from existing ones: take an irreducible polynomial f in F[x] and look at F[x]/(f), which is a new field of dimension deg f as a vector space over F; or take the field of fractions of F[x] to get F(x). Given a field extension F contained in K, the field generated over F by a single element alpha in K is of one of the two types.
• (2/13/15): Midterm
• (2/16/15): (No class, President's Day)
• Lecture 15 (2/18/15): (Section 13.2) Algebraic extensions. Given F in K in L, we have [L:F] = [L:K][K:F]. The set of algebraic elements in an extension F to K is a subfield of K. If K/F and L/K are algebraic extensions then L/F is algebraic.
• Lecture 16 (2/18/15): (Section 13.4) Algebraically closed fields. Every field F has an algebraic closure (proof using Zorn's lemma).
• Lecture 17 (2/20/15): (Section 13.4) Given a field F and an irreducible polynomial f, there exists an extension K of F in which f has a root. Given two roots alpha_1, alpha_2 of f there is an isomorphism F(alpha_1) to F(alpha_2) fixing F. Splitting fields. Lifting lemma: Given any isomorphism of fields F to F' taking a polynomial f to f', there is an isomorphism of a splitting field E of f over F to a splitting field E' of f' over F'.
• Lecture 18 (2/23/15): (Section 14.1) Automorphism groups of fields. Examples. Galois extensions. The splitting field of a separable polynomial is Galois.
• Lecture 19 (2/25/15): (Section 14.2) Characterizations of Galois extensions. If L is the fixed field of a finite subgroup of aut(K), then K/L is normal and separable.
• Lecture 20 (2/25/15): (Section 13.5) Separable and inseparable extensions. An irreducible polynomial is inseparable if and only if df/dx = 0; this happens only in characteristic p. Perfect fields. All irreducible polynomials over perfect fields are separable. Purely inseparable extensions.
• Lecture 21 (2/27/5): (Section 14.2, 14.4) A finite degree extension has a primitive element if and only if there are finitely many intermediate subfields. Any Galois extension has finitely many intermediate subfields. Any finite degree separable extension has a Galois closure and so has a primitive element.
• Lecture 22 (3/2/15): (Section 14.2) Finish proving all characterizations of Galois extensions are equivalent. If L is the fixed field of a finite subgroup H of aut(K), then K/L has degree |H| and H = Gal(K/L). Proof of the fundamental theorem of Galois theory.
• Lecture 23 (3/4/15): (Section 14.2) Examples of calculating Galois groups.
• Lecture 24 (3/4/15): (Section 13.6, 14.5) Cyclotomic polynomials. Proof that the nth Cyclotomic polynomial Phi_n(x) has coefficients in Z and is ireducible over Q. Proof that adjoining a root of x^n -1 to Q is a Galois extension with Galois group isomorphic to Z_{phi(n)}.
• Lecture 25 (3/6/15): (Section 14.3) Structure of finite fields: for each prime power p^n there is a unique field F_{p^n} with p^n elements, which can be constructed as the splitting field of x^{p^n} - 1 over F_p.
• Lecture 26 (3/9/15): (Section 14.7) If K = F(alpha) where alpha is a root of x^n -a and F has n distinct roots of x^n -1 already, then K/F is Galois with cyclic Galois group. Linear independence of characters. If K/F is a Galois extension with cyclic Galois group Z_n, then if x^n -1 splits with distinct roots in F then K = F(alpha) for some alpha which is a root of x^n - a for some a in F. Discussion of solvability by radicals.
• Lecture 27 (3/11/15): (Section 14.6, 14.7) The polynomial x^5 -6x + 3 is not solvable by radicals because its Galois group is S_5. Proof of the theorem that a rational polynomial f(x) is solvable by radicals if and only if the Galois group of its splitting field over Q is solvable. Proof of the fundamental theorem of algebra: C is algebraically closed.

Homework

• Homework 1: Homework 1 due Friday January 16

• Homework 2: Homework 2 due Friday January 30

• Homework 3: Homework 3 due Friday February 13

• Midterm Midterm exam given Friday February 13

• Homework 4: Homework 4 due Friday February 27

• Homework 5: Homework 5 due Friday March 13