Schedule of lectures

We will cover much of Chapters 10 and 12-14 of the text, but not always in the order the topics are presented in the text. The following schedule will be updated as we go with information about what we covered on particular days.

• 1/6/2020 (Section 10.1-10.2) Definition of a left R-module. Examples. Homomorphisms, submodules, factor modules, and examples. Fundamental homomorphism theorems for modules. Left R-modules are the same as right R^op modules. The set Hom_R(M,N) of module homorphisms from M to N is an Abelian group and is an R-module if R is commutative.
• 1/8/2020 (Section 10.1-10.2) End_R(M) = Hom_R(M,M) is a ring where the multiplication operation is composition, called the endomorphism ring of the module M. Examples: if V is an n dimensional vector space over F then End_F(V) is isomorphic to M_n(F). If R is a left R-module by multiplication then End_R(R) is isomorphic to R^op. An R-module structure on an Abelian group M is the same thing as a ring homomorphism R --> End_Z(M). An F[x]-module where F is a field is the same as an F-vector space V together with a choice of F-linear map V --> V which tells you how x acts. Definition of the direct sum and product of an indexed set of R-modules.
• 1/10/2020 (Section 10.3) The submodule generated by a subset. Finitely generated modules. Cyclic modules. Every cyclic left R-module is of the form R/I for a left ideal I. Free modules. The universal property of a free module. A direct sum of copies of R as a left module is free, and every free module is of this form. Any two free modules with bases of the same cardinality are isomorphic.

• 1/13/2020 (Section 12.1) Internal direct sums. Indecomposable modules. Examples. Rank of a module over an integral domain R. There are at most n R-linearly independent elements in R^n; in particular R^n has rank n. Statement of the main theorem classifying finitely generated modules over a PID (invariant factor form). Examples where it fails for infinitely generated modules or for finitely generated modules over non-PIDs. Torsion submodule of a module over an integral domain. Comments on how to prove the uniqueness part of the classification theorem.
• 1/15/2020 (Section 12.1) Proposition: If R is a PID, a submodule N of R^n is also free, and one can choose a basis y_1, ..., y_n of R^n such that a_1y_1, ..., a_ny_n is a basis of N, where each a_i divides a_{i+1}. Proof of the main theorem classifying f.g. modules over R using this proposition.
• 1/17/2020 (Section 12.1) The uniqueness part of the classification theorem of f.g. modules over PIDs (invariant factor form). The Elementary divisor form of the theorem. Examples of calculating the elementary divisors from the invariant factors and vice versa.

• 1/20/2020 MLK day (no class)
• 1/22/2020 (Section 12.2) Review of correspondence between linear transformations and matrices. Two matrices are similar if and only if they represent the same linear transformation with respect to two different bases. Given a linear transformation of a finite dimensional vector space $V$, there is a unique basis of V so that the corresponding matrix is in rational canonical form. Every matrix is similar to a unique rational canonical form.
• 1/24/2020 (Section 12.3) The Jordan canonical form. Every matrix is similar to a Jordan canonical form, which is unique up to permutation of the Jordan blocks. Generalized eigenspaces. Relation of the elementary divisors of a linear transformation to the generalized eigenspaces.

• 1/27/2020 (Section 12.3) The minimal polynomial of a matrix (or linear transformation). The characteristic polynomial. The minimal polynomial is equal to the largest invariant factor. The characteristic polynomial is the product of all of the invariant factors. Corollary: the Cayley-Hamilton theorem, every matrix satisfies its own characteristic polynomial. Examples of calculating rational and Jordan forms.
• 1/29/2020 (Section 12.3, 10.4) Matrices are similar over one field if and only if they are similar over an extension field (proof using rational form). Definition of R-balanced maps. Definition of the tensor product-a universal object for R-balanced maps. Existence of the tensor product (uniqueness is exercise).
• 1/31/2020 (Section 10.4) R otimes_R M is isomorphic to M. Z/n otimes_Z Q is 0. If R is commutative then M otimes_R N is an R-module again. If V and W are vector spaces over F then a basis for V otimes_F W is (v_i otimes w_j) where {v_i} and {w_j} are bases of V and W respectively. Extension of scalars.

• 2/3/2020 (Section 10.4, 10.5) More on extension of scalars. If F is a subfield of a field K, then for any F-vector space V, K otimes_F V is a K vector space and dim_F(V) = dim_K(K otimes_F V). Definition of an R-algebra for a commutative ring R. An R-algebra is the same as a ring S with a homomorphism from R to the center of S. If A and B are R-algebras, then A otimes_R B is again an R-algebra. Example: F[x] otimes_F F[y] is isomorphic to F[x, y] as F-algebras. Exactness and short exact sequences. flatness.
• 2/5/2020 (Section 10.4, 10.5) Given a short exact sequence 0 -> M -> N -> P -> 0, then for any module Q otimes_R M -> Q otimes_R N -> Q otimes_R P -> 0 is exact (right exactness of tensor). So Q is flat if and only if the functor (Q otimes_R -) preserves injections. Z/nZ is not a flat Z-module. Free modules are flat. Direct sums pull out of either coordinate of the tensor (but products do not). Projective modules. Projective modules are the same as direct summands of free modules (proof omitted). Projective modules are flat.
• 2/7/2020 (No class)

• 2/10/2020 Midterm Exam (in class)
• 2/12/2020 (Section 13.1) Field extensions. Degree of an extension [F:E]. Extensions generated by a subset. Any simple extension E(alpha) is either isomorphic to E(x) if x is transcendental, or to E[x]/(f) if alpha is algebraic over E, in which case the unique monic such f is irreducible and called the minimal polynomial over alpha over E. Examples.
• 2/14/2020 (Section 13.2) Algebraic extensions. If E \subseteq F \subseteq K then [K:E] = [K:F][F:E]. Finite degree extensions are algebraic. The set of algebraic elements in an extension is closed under all field operations and so is a subfield. If E \subseteq F \subseteq K with F/E and K/F algebraic, then K/E is algebraic.

• 2/17/2020 President's Day (No class)
• 2/19/2020 (Section 9.4, 13.6) Irreducible tests for polynomials. Rational root tests. Eisenstein criterion. Examples. Tricky use of Eisenstein. Cyclotomic polynomials. The nth cyclotomic polynomial is irreducible (omitted proof)
• 2/21/2020 (Section 13.4) Splitting fields. Any irreducible polynomial over a field has a root in some extension field. Definition of a splitting field. Splitting fields exist. Any two splitting fields of f over F are isomorphic, via an isomorphism that sends any root of an irreducible factor of f to any other root of that factor. Examples.

• 2/24/2020 (Section 13.5) Separable polynomials and separable extensions. f is separable if and only if gcd(f, f') = 1, where f' is the derivative of f. An inseparable irreducible polynomial f can only occur in characteristic p and must be of the form g(x^p). The frobenius map of a field of characteristic p. A perfect field is a field of characteristic 0 or one of characteristic p such that the Frobenius is surjective. All irreducible polynomials over a perfect field are separable. Example of an inseparable extension.
• 2/26/2020 (Section 13.5, 14.4) The multiplicative group of a finite field is cyclic. A finite field has prime power order p^n and is the splitting field of x^{p^n} of F_p. There is such a field for each prime power p^n and any two fields of size p^n are isomorphic. There are irreducible polynomials of degree n over F_p for all n. Theorem of the primitive element: a finite degree extension K/F is simple or has a primitive element (i.e. K = F(alpha) for some alpha) if and only if there are finitely many intermediate fields between F and K.
• 2/28/2020 (Section 14.1, 14.2) The automorphism group Aut(K/F). The extension K/F is called Galois if Aut(K/F) has the same number of elements as [K:F]. The splitting field of a separable polynomial is Galois. If H is a subgroup of Aut(K), Fix(H) is the set of elements of K fixed by all automorphisms in H. If H is a finite subgroup of Aut(K) then K/Fix(H) is an algebraic, separable, normal extension.

• 3/2/2020 (Section 14.2) Equivalent characterizations of an extension F subseteq K being Galois: (i) Aut(K/F) has the same size as [K:F]; (ii) the fixed field of Aut(K/F) is F; (iii) K/F is normal and Separable; (iv) K is the splitting field over F of a separable polynomial. Corollaries. If E is an intermediate field then K/E is Galois and fix(Aut(K/E) = E. A finite degree separable extension is contained in a Galois extension (Galois closure). Every finite degree separable extension has a primitive element. Fundamental Theorem of Galois theory, beginning of proof.
• 3/4/2020 (Section 14.2, 14.5) Conclusion of proof of the Fundamental Theorem. Cyclotomic extensions--- if zeta is a primitive nth root of 1, then Q(zeta)/Q is Galois with Galois group (Z/nZ)*. Specific example, n = 9, and finding all of the intermediate fields. Another example: The splitting field of x^4 -2 over Q.
• 3/6/2020 (Section 14.3, 14.5) Another example, the Galois group of the extension of Q generated by sqrt(2 + sqrt(2)). The Galois group of a finite field over its prime subfield is cyclic. Classification of subfields of finite fields. Beginning of Kummer theory: if F contains n distinct nth roots of 1 and K = F(alpha) where alpha is a root of x^n -a, then K/F is Galois with Galois group cyclic of order d dividing n.

• 3/9/2020 (Section 14.2, 14.7) Linear independence of characters. The converse of Kummer's theorem: If F contains n distinct nth roots of 1 and K/F is Galois with cyclic Galois group of order d, then K = F(alpha) where alpha is a root of x^d -a for some a in F. Root extensions and polynomials solvable by radicals. Galois's Theorem: in charactersitic 0, a polynomial f in F[x] is solvable by radicals if and only if Gal(K/F) is a solvable group, where K is the splitting field of f over F. Idea of the proof of Galois's theorem.
• 3/11/2020 (Section 14.7, 13.4) Example of a degree 5 polynomial which is not solvable by radicals over Q. Basics of algebraically closed fields and algebraic closures. Proof that any field has an algebraic closure.
• 3/13/2020 (Section 14.6) Galois-theoretic proof that C is algebraically closed. A fun theorem about C: the Artin-Schreier theorem.

• 3/18/2020 Final Exam 11:30am-2:30pm