MWF 11:3012:25 CL room 115, TR 11:3012:25 140 W 19th ave, room 136
Instructor: Sasha Leibman  office: MW406 office hours: Wednesday 23pm, Thursday 12pm 
email: leibman.1@math.osu.edu  
phone: 6142920663 
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(^{n}√a)/Q – they are of the form Q(^{d}√a)/Q for dn. 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(^{n}√a)/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, 12pm2pm, at CL 115, OR (up to your choice) on Tuesday, May 1, 10am12pm, at 136 in 140 W 19th ave building. The exam is open books and notes. It will consist of 67 problems, 3 of which will be related to modules, tensor products, the theory of moudles over PIDs, and normal forms of matrices, and 34 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 343345);
10.2.1, 2, 3, 7, 8 (page 350); 10.3.2, 3, 4, 5, 6, 7, 8, 9, 10 (pages 356358). 
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 356357).  
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 356358). 
written part:  due by Tuesday, January 23   Exercises 
10.2.6, 11 (page 350);
10.3.15, 22(ac) (pages 357358),
and the following problem:
Let M be an Rmodule 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 375377). 
written part:  due by Tuesday, January 30   Exercises  10.4.9, 10(a,b), 16(a,b), 20, 21(a,b) (pages 376377).  
HW4  recommended:  by Friday, February 2   Exercises 
10.5.19 (pages 403404);
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 422423);
11.3.1, 2, 4, 5 (page 435). 
written part:  due by Wednesday, February 14   Exercises  11.2.11(ac) (page 423); 11.3.3(af) (page 435).  
HW6  recommended:  by Friday, February 16   Exercises  11.5.1, 3, 4, 5, 7, 9, 13 (pages 454455). 
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 468472). 
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 488489);
12.3.1, 2, 5, 6, 9, 17, 18, 19, 20, 21, 22, 23, 24 (pages 499501). 
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 529531); 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 530531); 13.4.2 (page 545).  
HW10  recommended:  by Friday, March 30   Exercises 
13.5.6, 7, 8 (pages 551552);
13.6.1, 3, 4, 5 (pages 555556); 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 566567);
14.2.4, 6, 7, 8, 12, 16 (pages 581582); 14.3.10, 11 (page 589). 
written part:  due by Wednesday, April 11   Exercises  14.2.3, 10, 13, 14, 15(a,b) (pages 581582); 14.3.8 (page 589).  
HW12  recommended:  by Friday, April 20   Exercises 
13.3.1, 4, 5 (pages 535536);
14.2.28 (page 584); 14.4.1, 5 (pages 595596); 14.5.10 (page 603); 14.6.1, 2, 4, 5, 10, 11, 12, 17, 18, 19, 20 (pages 617618); 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. 10.3.911)
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 10.3.1617)
Internal direct sums of submodules (Section 10.3) The pprimary 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 contravariant 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 12.1.1619)
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 13.2.1920) 
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. 