# Math 200b Winter 2012

## Announcements:

### Instructors:

#### Professor:

Name Office E-mail Phone Office Hours Lecture Time Lecture Place
Prof. Daniel Rogalski AP&M 5131 drogalsk@math.ucsd.edu 534-4421 M 4-5pm, Tu 11am-12pm MWF 2-2:50pm AP&M 5402

#### Teaching Assistants:

Name Office E-mail Office Hours
James Berglund 6331 AP&M jberglun@math.ucsd.edu W 1-2pm, Th 1-2pm

Course description:

This is the second quarter of the three-part graduate algebra sequence. In this quarter, we will cover primarily module and field theory as in Chapters 10-14 of Dummit and Foote's text, Abstract Algebra. Please find the more detailed syllabus for the class, including information about homework, exams, academic honesty, etc., below:

### Math 200b course syllabus

Lecture Summaries

• Lecture 1 (01/09/12): (Section 10.1-10.2) Definition of a module. Examples: R as a module over itself, left ideal I, R/I, vector spaces over fields, etc. Z-modules are the same as abelian groups. Submodules and factor modules. homomorphisms of modules and isomorphism theorems.
• Lecture 2 (01/11/12): (Section 10.2) Hom_R(M,N) is an abelian group, and an R-module if R is commutative. End_R(M) = Hom_R(M,M) is a ring with product being composition of functions. End_R(R) is isomorphic to R as rings if R is commutative. There is a 1-1 correspondence between R-module structures on an abelian group M and ring homomorphisms R to End_Z(M) (proof will be an exercise). A module over F[x], where F is a field, is the same as an F-vector space V together with a linear transformation phi: V to V that tells you how x acts. Then F[x]-submodules corespond to subspaces which are phi-invariant.
• Lecture 3 (01/13/12): (Section 10.3) Direct sums and direct products of modules. The submodule generated by a subset. The definition of a module M being free on a subset X, and the universal property of free modules.
• NO CLASS (MLK day) (01/16/12)
• Lecture 4 (01/18/12): (Section 12.1) The fundamental theorem on f.g. modules over PIDs. Proof of existence part of the fundmental theorem, assuming technical result (Theorem 4 on p. 460 on text). Torsion and torsionfree modules.
• Lecture 5 (01/20/12): (Section 12.1) Sketch of uniqueness part of the fundamental theorem. Rank of a free module over a commutative ring is invariant.
• Lecture 6 (01/23/12): (Section 12.2) Similarity of matrices. Rational canonical form, derived using the invariant factor form of the fundamental theorem of f.g. modules over PIDs applied to a F-vector space V as a F[x]-module where x acts as some linear transformation phi: V to V.
• Lecture 7 (01/25/12): (Section 12.2) Characteristic and minimal polynomials of a matrix. Relationship to the rational canonical form. The Cayley-Hamilton theorem: any matrix satisfies its characteristic polynomial; equivalently, the minimal polynomial divides the characteristic polynomial. Also, every irreducible factor of the characteristic polynomial already divides the minimal polynomial. Another application: two matrices are similar over F if and only if they are similar over any extension field K of F.
• Lecture 8 (01/27/12): (Section 12.3) Jordan canonical form (over an algebraically closed field). Relation to generalized eigenspaces.
• Lecture 9 (01/30/12): (Section 10.4) Tensor products of modules (over a commutative ring only). The definition as a universal property. Uniqueness up to isomorphism. Construction of the tensor product. Some properties and examples.
• Lecture 10 (02/01/12): (Section 10.4) More on tensor products. The special case of tensor over a field K, where we have an explicit basis for V tensor_K W. The special case of base ring extension: S tensor_R M where R is a subring of S, makes an R-module M into an S-module.
• Lecture 11 (02/03/12): (Section 10.4.-10.5). K-algebras. The tensor product of two K-algebras is a K-algebra. Examples of tensor products of algebras. Short exact sequences. Equivalence of short exact sequences.
• Lecture 12 (02/03/12): (Section 10.5) Split short exact sequences and characterizations via splitting maps. For any R-module Q, the functor Q otimes_R -- is right exact. Applying this functor to a split exact sequence, it remains split.
• Lecture 13 (02/03/12): (Section 10.5) Hom. Left exactness of Hom_R(--, Q) and Hom_R(Q, --). Projective modules. Free modules are projective. A module is projective if and only if it is a direct summand of a free module.
• Lecture 14 (02/03/12): (Section 10.5) A module P is projective if and only if Hom(P, --) is exact. Injective modules. I is injective if and only if Hom(--, I) is exact. Flat modules. Projective modules are flat. Baer criterion for injective modules. Examples of injective modules: divisible modules over PIDs are injective.
• Midterm Exam (02/13/12):
• Lecture 15 (02/15/12): (Section 13.1) Field extensions. Given an extension E subset F, the field E(a_1, dots, a_n) generated by a set of elements. The structure of a simple extension E(a): either a is transcendental and this is isomorphic to E(x), or a is algebraic and this is isomorphic to E[x]/(g(x)) for some irreducible polynomial g in E[x].
• Lecture 16 (02/17/12): (Section 13.2) Algebraic extensions. Given a chain of extensions E subset F subset K, one has [K:E] = [F:E][K:F]. Application: sum, product, difference, quotient of algebraic elements is algebraic. The set of all elements of an extension E subset K which are algebraic over E is a subfield of K.
• Presidents' Day (NO CLASS) (02/20/12):
• Lecture 17 (02/22/12): (Section 13.4) Adjoining two different roots of an irreducible polynomial f \in F[x] to F yields isomorphic extensions. Definition of splitting fields. Any two splitting fields of f over F are isomorphic.
• Lecture 18 (02/24/12): (Section 13.4, 14.1) Examples of Splitting fields: splitting fields of x^n - 1 and x^p - 2 over Q (p prime), and their degrees over Q. Automorphisms of fields and Aut(K/F). An extension is called Galois if |Aut(K/F)| = [K:F]. The splitting field of a separable polynomial is Galois.
• Lecture 19 (02/27/12): (Section 14.2) 4 characterizations of an extension K/F being Galois: |Aut(K/F)| = [K:F]; fix(Aut(K/F)) = F; K/F is separable and normal; and K is the splitting field of a separable polynomial over F. Corollaries: |Aut(K/F)| is at most as big as [K:F] for any finite extension.
• Lecture 20 (02/29/12): (Section 14.2) If H is a subgroup of Aut(K/F) and L = fix(H), then Aut(K/L) = H (proof assuming the theorem of the primitive element.) The fundamental theorem of Galois theory. Example of correspondence between fields and subgroups for the splitting field of x^3 - 2 over Q.
• Lecture 21 (03/02/12): (Section 13.5) Separable polynomials. f is separable if and only if gcd(f,f')=1. If f is irreducible and f' is not zero, then f is separable; in particular this holds in characteristic 0. If f is irreducible in characteristic p, there exists an irreducible and separable polynomial g such that f(x)=g(x^{p^k}). Examples.
• Lecture 22 (03/05/12): (Section 13.5-13.6) Basics of finite fields, including their Galois group. Perfect fields. Cyclotomic polynomials \Phi_n(x). x^n-1=\prod_{d|n} \Phi_d(x).
• Lecture 23 (03/07/12): (Section 14.2, 14.3) An example of a Galois group: splitting field over Q of sqrt(2 + sqrt(2)). The intermediate fields between F_p and F_{p^n} are in one to one correspondence with the divisors of n. x^(p^n) - x is the product of all irreducibles over F_p of degree dividing n.
• Lecture 24 (03/09/12): (Section 14.4, 14.5, 14.7) The theorem of the primitive element: K/F is a simple extension if and only if there are finitely many intermediate fields between F and K. Cyclotomic extensions. The galois group of Q(zeta)/Q, where zeta is a primitive nth root of 1, is isomorphic to the units group of Z/nZ. Gal(F(\alpha)/F) is cyclic when alpha is a root of x^n - a, when F already contains n distinct nth roots of 1.
• Lecture 25 (03/12/12): (Section 14.7) Linear independence of characters. If K/F is a Galois extension where F contains n distinct roots of 1, and Gal(K/F) is cyclic of order n, then K = F(alpha) where alpha is a root of x^n - a some a in F. Root extensions. Statement of Galois theorem: F of char. 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. Example of a degree 5 polynomial in Q[x] which is not solvable by radicals.
• Lecture 26 (03/14/12): (Section 14.7) Proof of Galois theorem.
• Lecture 27 (03/16/12): (Section 13.4, 14.6) Every field has an algebraic closure (proof using trick of Artin). Galois-theoretic proof of fundamental theorem of algebra that C is algebraically closed.

Homework