### Math 5591H and 5112, Algebra II

MWF 11:30-12:25 CL room 115, TR 11:30-12:25 140 W 19th ave, room 136

 Instructor: Sasha Leibman office: MW406     office hours: Wednesday 2-3pm, Thursday 1-2pm e-mail: leibman.1@math.osu.edu phone: 614-292-0663

Textbook: D.S.Summit and R.M.Foote, Abstract Algebra, 3rd edition

Handouts: A proof of the main part of the Galois theorem.
Galois groups of cubics and quartics, by Keith Conrad.

As for the problem that I was unable to resolve in class, the answer is easy: We know all subextensions of Q(na)/Q – they are of the form Q(da)/Q for d|n. None of these extensions is normal, except for d=2, which is not possible when n is odd. However, all subextensions of the abelian extension Q(ω)/Q are normal, so the extensions Q(na)/Q and Q(ω)/Q have a trivial intersection for any odd n. (Was this what Ryan said in class?)

Midterm 1 on Wednesday, February 7. Solutions. List of topics. Sample problems.
Midterm 2 on Tuesday, March 6. Solutions. List of topics. Sample problems.
Midterm 3 on Tuesday, April 3. Solutions. List of topics. Sample problems. Solution to problems 18 and 3 from the list.

Final exam is going to be on Friday, April 27, 12pm-2pm, at CL 115, OR (up to your choice) on Tuesday, May 1, 10am-12pm, at 136 in 140 W 19th ave building. The exam is open books and notes. It will consist of 6-7 problems, 3 of which will be related to modules, tensor products, the theory of moudles over PIDs, and normal forms of matrices, and 3-4 will be devoted to fields and the Galois theory. Review problems on Galois theory.

Homework:
 HW1 recommended: by Friday, January 12 -- Exercises 10.1.1, 2, 3, 5, 6, 7, 11, 12, 13, 18, 19, 20, 22 (pages 343-345); 10.2.1, 2, 3, 7, 8 (page 350); 10.3.2, 3, 4, 5, 6, 7, 8, 9, 10 (pages 356-358). written part: due by Tuesday, January 16 -- Exercises 10.1.8(a,b,c), 9, 10 (page 344); 10.2.9 (page 350); 10.3.7, 12(a,b) (pages 356-357). HW2 recommended: by Friday, January 19 -- Exercises 10.2.4, 5, 10, 12 (page 350); 10.3.13, 14, 20, 21, 23, 24, 27 (pages 356-358). written part: due by Tuesday, January 23 -- Exercises 10.2.6, 11 (page 350); 10.3.15, 22(a-c) (pages 357-358), and the following problem: Let M be an R-module and let I, J be ideals in R. (a) Prove that Ann(I+J)=Ann(I)∩Ann(J). (b) Prove that Ann(I∩J)⊇Ann(I)+Ann(J). (c) Give an example where the inclusion in (b) is strict. (d) If R is commutative and unital and I, J are comaximal, prove that Ann(I∩J)=Ann(I)+Ann(J). HW3 recommended: by Friday, January 26 -- Exercises 10.4.2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 18, 19, 24, 25 (pages 375-377). written part: due by Tuesday, January 30 -- Exercises 10.4.9, 10(a,b), 16(a,b), 20, 21(a,b) (pages 376-377). HW4 recommended: by Friday, February 2 -- Exercises 10.5.1-9 (pages 403-404); 11.5.4, 6, 8 (page 455). No written part. HW5 recommended: by Monday, February 12 -- Exercises 11.2.1, 2, 3, 5, 6, 7, 9, 10 (pages 422-423); 11.3.1, 2, 4, 5 (page 435). written part: due by Wednesday, February 14 -- Exercises 11.2.11(a-c) (page 423); 11.3.3(a-f) (page 435). HW6 recommended: by Friday, February 16 -- Exercises 11.5.1, 3, 4, 5, 7, 9, 13 (pages 454-455). written part: due by Tuesday, February 20 -- Exercises 11.5.5, 12(a,b,c) (page 455); 11.4.2, 3 (page 441). HW7 recommended: by Friday, February 23 -- Exercises 12.1.1, 2, 3, 5, 6, 8, 10, 13, 14, 20, 21, 22 (pages 468-472). written part: due by Tuesday, February 27 -- Exercises 12.1.4, 9, 11, 12(a,b) (page 469). HW8 recommended: by Friday, March 2 -- Exercises 12.2.3, 4, 5, 7, 10, 11, 12, 14, 15, 16, 17, 18 (pages 488-489); 12.3.1, 2, 5, 6, 9, 17, 18, 19, 20, 21, 22, 23, 24 (pages 499-501). No written part. HW9 recommended: by Friday, March 23 -- Exercises 13.1.1, 2, 4, 5 (page 519); 13.2.3, 4, 5, 6, 7, 10, 12, 14, 19, 21 (pages 529-531); 13.4.1, 3 (page 545). written part: due by Tuesday, March 27 -- Exercises 13.1.3 (page 519); 13.2.8, 9, 13, 16, 20 (pages 530-531); 13.4.2 (page 545). HW10 recommended: by Friday, March 30 -- Exercises 13.5.6, 7, 8 (pages 551-552); 13.6.1, 3, 4, 5 (pages 555-556); 14.3.1, 2, 3, 7 (page 589). written part: due by Tuesday, April 3 -- Exercises 13.5.5 (page 551); 13.6.6 (page 555); 14.3.4 (page 589). HW11 recommended: by Friday, April 6 -- Exercises 14.1.1, 4, 5, 6, 7, 8 (pages 566-567); 14.2.4, 6, 7, 8, 12, 16 (pages 581-582); 14.3.10, 11 (page 589). written part: due by Wednesday, April 11 -- Exercises 14.2.3, 10, 13, 14, 15(a,b) (pages 581-582); 14.3.8 (page 589). HW12 recommended: by Friday, April 20 -- Exercises 13.3.1, 4, 5 (pages 535-536); 14.2.28 (page 584); 14.4.1, 5 (pages 595-596); 14.5.10 (page 603); 14.6.1, 2, 4, 5, 10, 11, 12, 17, 18, 19, 20 (pages 617-618); 14.7.3, 4, 5, 6, 7, 8 (page 636). No written part.

My solutions to HW1, HW2, HW3, HW5, HW6, HW7, HW9, HW10, HW11
Exercise 14.2.7
Some problems from section 14.6.

Studied topics:
 January 8: Definition and examples of modules (Section 10.1) January 9: Homomorphisms of modules (Section 10.2) January 10: Simple modules. Schur's lemma (ex. 10.3.9-11) Factorization of modules (Section 10.2) Isomorphism theorems for modules (Section 10.2) Generating sets of modules (Section 10.3) January 11: The universal property of free modules (Section 10.3) Direct products and sums of modules. The universal property of direct products and sums (Section 10.3) January 12: Exercises from Section 10.1. January 16: Infinite direct products and sums and their universal prperties. January 17: The Chinese remainder theorem for modules (execises 10.3.16-17) Internal direct sums of submodules (Section 10.3) The p-primary components of a module (execises 10.3.18,22) Free modules (Section 10.3) January 18: "Internal" free modules and bases (Section 10.3) Vector spaces are free modules and their rank is well defined (Section 11.1) January 19: Exercises from Section 10.3. January 22: Maximal linear independent subsets in general modules (exercise 12.1.2) Definition of the tensor product of modules over a commutative ring (Section 10.4) January 23: Properties of tensor product (Section 10.4) Tensor product of direct sums (Section 10.4) January 24: Tensor product of a module and of a quotient of the ring (Section 10.4) Tensor product of a module and of the field of fractions (Section 10.4) January 25: Extension of scalars (Section 10.4) Tensor product of algebras (Section 10.4) Tensor product of homomorphisms (Section 10.4) January 26: Exercises from Section 10.4. January 29: The tensor algebra of a module. The symmetric and the exterior algebras (Section 11.5) January 30: Commutative diagrams and exact sequences of modules (Section 10.5) The short five lemma (Section 10.5) January 31: Flat modules (Section 10.5) February 1: Projective modules (Section 10.5) February 2: Injective modules (Section 10.5) February 5: Review. February 6: Review. February 7: Midterm 1. February 8: Homomorphisms of free modules of finite rank. Matrices of homomorphisms (Section 11.2) Change of basis and transition matrices (Section 11.2) February 9: Change of matrices under a basis change. Similar matrices. (Section 11.2) Dual modules, dual bases, and dual homomorphisms (Section 11.3) February 12: Finite dimensional vector spaces (Section 11.2) Exercises from Section 11.2. February 13: Exercises from Sections 11.2, 11.3. February 14: Contraction of tensors. The trace. Homomorphisms as tensors. Matrix multiplication. Co- and contra-variant tensors. February 15: The symmetric tensor algebra and the exterior algebra of a free module (Section 11.5) February 16: Symmetric and alternating tensors (Section 11.5) The determinant of a linear transformation (Section 11.4) February 19: Symmetric and alternating tensors (Section 11.5) February 20: Properties of the determinant of a linear transformation (Section 11.4) "Canonical forms" of a matrix of a linear transformation. The rank of a module over an integral domain (Section 12.1) February 21: Submodules of free modules of finite rank over a PID (Section 12.1) February 22: Finding bases of submodules of free modules of finite rank over EDs (Exercises 12.1.16-19) The fundamental theorem of modules of finite rank over PIDs. Invariant factors. (Section 12.1) February 23: Exercises from Section 12.1. Elementary divisors of modules of finite rank over PIDs (Section 12.1) February 26: Uniqueness of invariant factors (Section 12.1) Finding the invariant factors; the relations matrix (Section 12.1) February 27: The rational canonical form of the matrix of a linear transformation of a finite dimensional vector space (Section 12.2) February 28: The minimal polynomial of a transformation (Section 12.2) The Smith normal form of a matrix (Section 12.2) March 1: The characteristic polynomials of a transformation (Section 12.2) The Jordan canonical form of a matrix (Section 12.3) March 2: Review. March 5: Review. Exercises from Sections 12.2, 12.3. March 6: Midterm 2. March 7: Overview of the Galois theory. The prime subfield and the characteristic of a field (Section 13.1) March 8: Extensions of a field. Finite extensions, the degree of an extension (Sections 13.1, 13.2) Simple extensions. Algebraic and transcendental elements of an extension. The minimal polynomial of an algebraic element (Section 13.1) March 9: The structure and the degree of finite extensions (Section 13.2) Composites of finite extensions (Section 13.2) March 19: Examples of composites and towers of finite extensions. Algebraic extensions. Towers and composites of algebraic extensions are algebraic (Section 13.2) March 20: Maximal algebraic subextensions (Section 13.2) Finding the minimal polynomial of an algebraic element (see Exercises 13.2.19-20) March 21: Adjoining a root of an irreducible polynomial (Section 13.2) The splitting field of a polynomial (Section 13.4) March 22: The algebraic closure of a field (Section 13.4) March 23: Exercises from Sections 13.1,2,4. March 26: Normal extensions (see Exercises 13.4.5,6). March 27: Separable and inseparable polynomials, elements, and extensions. The Frobenius endomorphism. Perfect fields (Section 13.5) Finite fields (Sections 13.5, 14.3) March 28: Finite fields (Sections 13.5, 14.3) Roots of unity and cyclotomic extensions (Section 13.6) March 28: Cyclotomic extensions and cyclotomic polynomials (Section 13.6) March 29: Embeddings of algebraic extensions. Conjugate extensions (Sections 14.1,2) April 2: Review. April 3: Midterm 3. April 4: Galois extensions and Galois groups (Section 14.1) April 5: The fundamental Galois theorem – short version (Section 14.2) April 6: The fundamental Galois theorem – full version (Section 14.3) April 9: Exercises from Sections 14.1, 14.2. April 10: Exercises from Section 14.2. The composite of two extensions of which one is Galois (Section 14.4) April 11: A linear independence of radicals. The Galois group of the composite of two Galois extensions is a direct product (Section 14.4) The theorem on the primitive element (Section 14.4) April 12: Abelian extensions (Section 14.5) The fundamental theorem of algebra (Section 14.6) April 13: Constructions with ruler and compass (Sections 13.3, 14.5) April 16: Subfields of the field Q(n√a), with a>0. Polyquadratic extensions and constructible numbers (Sections 13.3, 14.5) April 17: The theory of symmetric rational functions (Section 14.6) Radical extensions and cyclic extensions (Section 14.7) April 18: Solvability of polynomials in radicals (Section 14.7) April 19: Solvability of polynomials in radicals (Section 14.7) The discriminant of a polynomial (Section 14.6) April 20: The Galois groups and solution in radicals of cubics and quartics (Sections 14.6, 14.7) April 23: Review problems.

Plans for the nearest future:
 Final exam – on April 27 or May 1, of your choice.