Instructors: Eran Assaf

Course on canvas.dartmouth.edu.

Syllabus

Week Date Topics Reading Homework
1 Wed 04 Jan Download Wed 04 Jan History of solving polynomial equations. The complex numbers.
Fri 06 Jan Download Fri 06 Jan Review of ring theory I: ring homomorphisms, ideals, fields and integral domains, polynomial rings.  DF 7.1-7.3
FT 7-10
2 Mon 09 Jan Download Mon 09 Jan Review of ring theory II: properties of ideals, characteristic, Euclidean domains, Principal Ideal Domains. 

DF 7.4,8.1-8.2

FT 10-11

Problem Set #1
Tue 10 Jan Download Tue 10 Jan (X-Hour) X-hour lecture: Noetherian domains, Unique Factorization Domains. Fundamental Theorem of Arithmetic. Reduction mod p. DF 8.3,9.1-9.2, 9.4
FT 11-12
Wed 11 Jan No class. 
Fri 13 Jan Download Fri 13 Jan Roots of polynomials. Irreducibility criteria for polynomials. Eisenstein's criterion, Gauss's Lemma. Factorization of polynomials. DF 7.5, 9.3-9.4
FT 12-14
3 Mon 16 Jan No class: Martin Luther King, Jr. Day Problem Set #2
Tue 17 Jan Download Tue 17 Jan (X-Hour) X-hour lecture:Field extensions. Simple extensions. The Tower Law for degrees. DF 13.1-13.2 FT 14-15
Wed 18 Jan Download Wed 18 Jan Algebraic extensions. Finitely generated extensions.Classification of finite extensions. Transcendental and algebraic elements. Minimal polynomial. Composite field.

DF 13.2

FT 15-21

Fri 20 Jan Download Fri 20 Jan Composite fields, continued. Compass and straightedge. Constructible numbers. Construction impossibility proofs. Regular n-gons. (A note on transcendental numbers Download A note on transcendental numbers)

DF 13.3
FT 22-24

4 Mon 23 Jan Download Mon 23 Jan Splitting fields. DF 13.4 
FT 28-29
Problem Set #3
Wed 25 Jan Separability. DF 13.5
FT 30-33
Fri 27 Jan Multiple roots. Embeddings. DF 13.4-13.5 
FT 27-31
5 Mon 30 Jan Field automorphisms. Automorphism group. DF 14.1 
FT 35-36
Problem Set #4
Tue 31 Jan (X-Hour) X-hour lecture: Algebraic closure.  DF 13.4 
FT 87-91
Wed 01 Feb Constructing automorphisms. Automorphism group calculations. Fixed fields.  DF 14.1 
FT 35-36
Fri 03 Feb Galois extensions. Linear independence of embeddings. DF 14.1-14.2 
FT 36-39, 67-68
5 Mon 06 Feb Fundamental theorem of Galois theory. Examples of the Galois correspondence. DF 14.2 
FT 39-43
Problem Set #5
Wed 08 Feb End of proof of the Galois correspondence. DF 14.2 
FT 39-43
Fri 10 Feb Normality. Finite fields. Cyclotomic fields. DF 14.2-3 
FT 53-55, 64-67
6 Mon 13 Feb Examples of Galois extensions. Computing Galois groups. DF 14.2 
FT 47-49
Midterm Exam
Wed 15 Feb Applications of the Galois correspondence. Radical extensions. Solvability by radicals. FT 44-46
Fri 17 Feb Radical extensions. Solvability by radicals. Galois's solvability theorem. DF 14.7 
FT 44-46,76-81
7 Mon 20 Feb Solvability, continued. Primitive element theorem. DF 14.7 
FT 59-61, 74-75
Problem Set #6
Wed 22 Feb Discriminant. Galois perspective on quadratic and cubic extensions. Quartic extensions. DF 14.7 
FT 49-51
Fri 24 Feb Quartic extensions.  DF 14.7 
FT 50-58 
8 Mon 27 Feb Finite fields. Cyclotomic fields.  DF 14.5 
FT 62-67
Problem Set #7
Wed 01 Mar Infinite Galois theory. Krull topology. Profinite groups. DF 650-652 
FT 93-103
Fri 03 Mar Fundamental Theorem revisited.  DF 650-652 
FT 93-103
9 Mon 06 Mar Taussky-Todd's theorem. Grace Hopper's thesis. Notices article Links to an external site.