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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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Fri 27 Jan | Multiple roots. Embeddings. | DF 13.4-13.5 FT 27-31 |
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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 |
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Wed 01 Feb | Constructing automorphisms. Automorphism group calculations. Fixed fields. | DF 14.1 FT 35-36 |
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Fri 03 Feb | Galois extensions. Linear independence of embeddings. | DF 14.1-14.2 FT 36-39, 67-68 |
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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 |
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Fri 10 Feb | Normality. Finite fields. Cyclotomic fields. | DF 14.2-3 FT 53-55, 64-67 |
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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 |
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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 |
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Fri 24 Feb | Quartic extensions. | DF 14.7 FT 50-58 |
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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 |
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Fri 03 Mar | Fundamental Theorem revisited. | DF 650-652 FT 93-103 |
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9 | Mon 06 Mar | Taussky-Todd's theorem. Grace Hopper's thesis. | Notices article Links to an external site. |