Calendar
| Week | Monday | Wednesday | Friday |
|---|---|---|---|
| 1 |
Group Actions, Orbits, Stabilizers Ch 14.1. Make sure you understand Theorem 14.11. Ch 5.4 Q32b. |
Class Equations (counting with orbits and stabilizers), Conjugacy Class Ch 14.2 (you can skip Corollary 14.16). |
Rings Ch 16.1, up to Prop 16.8. Give exmaples and counter-examples to each object in Figure 16.1. Also make sure you know what a zero divisor is. |
| 2 | Integral Domains and Fields Ch 16.2. Reading question 2,3. Ch 16.7 Q3ad. |
Check 1; Ring homomorphisms and Ideals Ch 16.3, up to Remark 16.28. Prop 16.27 is important, so you have to read the proof. Ch 16.7 Q6. |
Quotient Rings, Isomorphism Theorems Ch 16.3, Thm 16.29 - end. Read the proof for 16.30. Focus on Theorems 16.30, 16.31 and 16.34. |
| 3 | Maximal and Prime Ideals Ch 16.4. Pay attention to 16.35 and 16.38, which are super important. Ch 16.7 Q4a |
Check 2; Chinese Remainder Theorem Ch 16.5, up to Example 16.44. Focus on the examples. |
Polynomial Rings Ch 17.1. Theorem 17.5 is super important. |
| 4 | Catch-up / Review | Midterm 1 Solution | Division Algorithm, Euclidean Algorithm Ch 17.2. Work on Ch 17.5 Q3a, 4a. |
| 5 | Irreducible Polynomials Ch 17.3. Make sure you understand Theorem 17.14 and 17.17. |
Check 3; Field of Fractions Ch 18.1. Make sure you understand 18.5 - 18.7. |
UFD and PID Ch 18.2, up to Example 18.17. What is an example of a non-PID? |
| 6 | Polynomial Rings over Integral Domains Ch 18.2, ''Factorization in D[x]'' to end. Focus on ''primitive''. Give an example of a primitive polynomial, and a non-primitive polynomial, both in Z[x]. |
Check 4; Field extensions Ch 21.1, up to Example 21.6. Focus on examples. |
Algebraic extensions Ch 21.1, from ''Algebraic elements'' up to Theorem 21.9. |
| 7 | Catch-up / Review | Midterm 2 Solution | Minimal Polynomial Ch 21.1, from Theorem 21.10 to Example 21.14. Make sure to understand Examples 21.11 and 21.14. |
| 8 | Finite extensions Ch 21.1, from right after Example 21.14 to Theorem 21.22. Focus on Examples 21.20 and 21.21. |
Check 5; Algebraic Closures Ch 21.1, ''Algebraic Closures''. The most important part from this section is to understand the statement ''C is algebraically closed''. |
Splitting Fields Ch 21.2. Make sure to understand why Example 21.29 is a splitting field, but Exmaple 21.30 is not. You can skip Lemma 21.32 and Theorem 21.34, but you should read the statement of 21.36. |
| 9 | Memorial | Check 6; Finite Fields Ch 22.1. Make sure to understand Examples 22.8 and 22.13. Also pay attention to the statement of 22.1, 22.2, 22.5, 22.6, 22.10. |
Field Automorphisms Ch 23.1, up to Example 23.11. Focus on the examples. WARNING: The book calls the group ''Galois group'', denoted by G(E/F). In most other literature, this is NOT the Galois group. Instead, it is called ''Automorphism gropu of E over F'', denoted Aut(E/F). We will use this Aut(E/F) notation. The notion ''Galois group'' is reserved for next week, when the extension is Galois. |
| 10 | Separable extensions Ch 23.1, ''Separable Extensions''. Also re-read Lemma 22.5 from Ch 22.1. |
Check 7; Normal extensions Ch 23.2, up to Figure 23.22. Give an example of a normal extension and a non-normal extension of Q. |
Fundamental Theorem of Galois Theory Ch 23.2, from Thm 23.23 to end. Make sure to understand Example 23.25. |
| 11 | Final Exam: Jun 14 (Wed), 3-5pm (duration tentative) |
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