| Week |
Topic |
Assignments |
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1
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Review & Polynomial Rings |
Monday 1/10: Meeting via Zoom, Review of AA I: Day 1 Board Notes
Wednesday 1/12: Meeting via Zoom, Ch. 14 & 15 Review: Day 2 Board Notes & Discussion Notes
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Perusall Assignment: Syllabus & Research Project (due Monday 1/10)
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Review Poster Rough Draft: (due Tuesday 1/18)
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Homework 1 (due Wednesday 1/19) : LaTeX the following problems and turn in through WISE
- Ch.14/ #10-11, 51, 60
- Ch. 15/ #28, 44, 64
- Read and think about (but do not turn-in) the following problems: Ch. 15/ 22, 46, 47, 50, 54, 58, 66
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Sign-up for research group topics by Tuesday 1/18
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2
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Polynomial Rings & Factorization |
Monday 1/17: No Class - MLK Day
Wednesday 1/19: Meeting via Zoom, Ch. 16 Board Notes & Discussion Notes
Class Prep: Think about what makes the integers special. What are some of the main things you know about integers? Review definitions of polynomial degree, adding and multiplying polynomials, etc. What does it mean for two polynomials to be equal? Review polynomial long division. If your long division of integers is rusty, check out this video. Once you feel ok with long division generally, this video on polynomial long division might be a good resource. [Videos chosen for their balance of length and content, not for entertainment value or good match with audience - sorry. I recommend 1.75 speed ;) ]
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3
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Z and F[x] - What is the essence of integerness? |
Monday 1/24: meeting via Zoom, Ch. 17 - Polynomial Rings and Z Week #3 Board Notes and Discussion Notes
Wednesday 1/26: meeting via Zoom, Ch. 17 - Polynomial Rings and Reducibility Tests Week #3 Board Notes and Discussion Notes
- Watch Integral Domain Structure Videos by Monday 1/24:
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4
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Vector Spaces, Modules, Algebras |
Monday 1/31: meeting IN PERSON :), Ch. 19 - Vector Spaces, Modules, and Algebras, Class Notes
Wednesday 2/2: meeting IN PERSON :), Ch. 19 - Vector Spaces, Modules, and Algebras, Class Notes
- Homework 2 (due Monday 1/31) : LaTeX and turn in through WISE
- Perusall Assignment: Unit Conjecture - Quanta (due Wednesday 2/2)
- Homework 3 (due Monday 2/7): LaTeX the following problems and turn in during class or to my office
- Ch. 17/ #5(b or c), 19, 31
- Problem 4: We framed irreducibility tests in terms of the polynomials over Q and Z. Can any of these tests be generalized to the setting of a polynomials over an integral domain D and the associated field of quotients F? If not, can they be generalized by placing certain requirements on the domain D? Try to generalize the irreducibility tests, clearly stating the test in the more general setting (when possible) and specifying any requirements on D.
- Ch. 19/ #16
- Problem 6: Prove that if R is a commutative ring, the polynomial ring R[x] is an R-algebra. What is a basis for this algebra?
- Research Journal: In 2-5 sentences, summarize what the topic you are looking into is about. Write down the important definitions, concepts, and theorems you've come across so far and identify at least two questions you'd like answers to.
- Read and think about (but do not turn-in) the following problems: Ch. 17/ 3, 6, 10, 12, 15, 33, 34 and Ch. 19/ 13, 14, 15, 27, 31, 32. Given a ring R and an ideal I, prove I is an R-algebra. Prove every Abelain group is a Z-module.
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5
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Vector Spaces, Modules, Algebras
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Monday 2/7: Ch.19 Vector Spaces, Modules, & Algebras - examples
Wednesday 2/9: No Class - Use time to work on research journals or past assignments
- Week #5 Survey: https://forms.gle/AfhXMzjZfU4vVJ7g8.
- Homework 3 (due Monday 2/7): LaTeX and turn in to me directly or via WISE
- Perusall Assignment: Finite Fields & Parker Square - Quanta (due Wednesday 2/9)
- Research Journals: due Friday 2/11
- Summarize what the topic you are looking into is about. Write down the important definitions, concepts, and theorems you've come across so far and identify at least two questions you'd like answers to.
- List the resources you have used so far
- Make note of any connections to concepts we've already covered in class (this semester or last)
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6
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Extension Fields and Celebration of Knowledge |
Monday 2/14: meeting IN PERSON :), Ch. 20 - Extension Fields
Wednesday 2/16: meeting IN PERSON :), Celebration of Knowledge (study guide available on WISE under Resources > Handouts)
- Individual Celebration of Knowledge (in-class Wednesday 2/16 take-home due
Friday 2/18 Sunday 2/20)
- Perusall Assignment (2nd Chance): Finite Fields & Parker Square - Quanta (due
Saturday 2/19 Saturday 2/26)
- Homework 4 (due Wednesday, 2/23): LaTeX the following problems and turn in to me directly or via WISE
- Problem 1: clearly state and prove the 1st Isomorphism Theorem for vector spaces.
- i.e. given a homomorphism between F-vector spaces (aka a linear transformation), define a new map between related spaces and prove this new map is a vector space isomorphism (aka "a bijective linear transformation)
- Problem 2: Prove that given an F-vector space V and a subspace S of V, dim(V/S)=dim(V)-dim(S).
- It might help to recall the theorem from Linear Algebra that given a basis $$\{\vec{u_1},\vec{u_2},\ldots,\vec{u_k}\}$$ for the subspace S of V, you can find a basis for V of the form $$\{\vec{u_1},\vec{u_2},\ldots,\vec{u_k},\vec{v_{k+1}},\vec{v_{k+2}},\ldots,\vec{v_n}\}.$$
- Problem 3: Using the 1st isomorphism theorem for vector spaces and your result for Problem 2, prove the following two linear algebra theorems as succinctly as possible:
- a. Rank-nullity theorem
- If the matrix equation Ax=b has a solution xp, then every solution is of the form xp+x0 where x0 is a solution to the corresponding homogeneous equation.
- Perusall Assignment: Classifying Torsion Free Abelian Groups - Quanta (due
Friday 2/25 Saturday 3/3)
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7
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Extension Fields and Algebraic Extensions |
Individual Celebration of Knowledge (take-home due (upload to WISE) Sunday 2/20)
Monday 2/21: Mid-semester break, no class
Wednesday 2/23: meeting IN PERSON :), Ch. 27 & 29 Pinter Chapter - Extension Fields & Algebraic Extensions
- Homework 4 (due Wednesday, 2/23): LaTeX and turn in to me directly or via WISE
- Perusall Assignment (2nd Chance): Finite Fields & Parker Square - Quanta (due Saturday 2/26)
- Homework 5:
- Read proof of the Basic Theorem of Field Extension (Pinter pg 275) by Monday 2/28
- Prepare to present the following problems in class Monday 2/28
- Pinter Ch. 27/ A1(d)(g), B4(b), B5(c), D2, D3, D8, H1, H2
- prop 1: Given an irreducible polynomial p(x) in F[x], if char(F)=0, then p(x) has no multiple zeros. If char(F)=p, then p(x) has a multiple zero if and only if p(x)=g(x^p) for some g(x) in F[x].
- prop 2: Given E is an extension field of F, automorphisms that act as the identity on F map roots of irreducible polynomials in F[x] to roots of the same polynomial.
- LaTeX and turn in your solutions to D2, H2, prop. 2 on by Friday 3/4
- Take-home celebration problem 1 & 2 rewrites due Friday 3/4
- Perusall Assignment: Classifying Torsion Free Abelian Groups - Quanta (due Saturday 3/5)
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8
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Extension Fields and Algebraic Extensions |
Monday 2/28: Discussion of Homework #5 Problems
Wednesday 3/2: (Pinter Ch. 29 / Gallian Ch. 21) - Extension fields from a vector space point of view, Discussion of Homework #5 Problems
- Homework 6:
- Read Pinter Perusall assignment (Ch. 29 through Thm 2) by Monday 3/7 (Extra credit if you read and comment through Perusall)
- Prepare to lead discussion of following problems in class Monday 3/7 - Pinter Ch. 29/ A4, A6, B1, B4, D3, D4
- Prepare to lead discussion of following problems in class Wednesday 3/9 - Pinter Ch. 29/ D6, E1, F1, F2, F5, G2
- LaTeX and turn in your solutions to Pinter Ch. 29/ D4, D6, F2, G2 by Friday 3/11
Research Journal Collection Friday 3/11
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9
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Extension Fields and Algebraic Extensions |
Week 9 Survey (Masks & Research Journal): https://forms.gle/kSybQ7k4eh4Sa3Xs5
Monday 3/7: Pinter (Ch. 29)/Gallian (Ch.20) - Field Extensions as Vector Spaces, Discussion of Homework #6 Problems
Wednesday 3/9: Pinter (Ch. 31 & 32) - Pinter Chapter Galois Theory, Discussion of Homework #6 Problems
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10
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Homomorphisms between Extension Fields & Group Celebration of Knowledge |
Monday 3/14: Pi Day!!!! Come Celebrate in the math hearth at 4:15, but first....
... we have an exciting class applying homomorphisms to field Extensions! Pinter Ch. 31 here we come! There are not assigned problems to present this week, but be prepared to collaborate on problems in class
Wednesday 3/16: Group Celebration of Knowledge - Vector Spaces to Extension Fields and everything in between
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11
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---,-'-(# Monday 3/21-Friday 3/25 Spring Break #)-'-,---
(My attempt at making a flower)
2nd Chance Perusall: I will open the Perusall due dates back up over spring break, so if you've missed any, now's your chance to get caught up.
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| 12 |
Galois Theory |
Monday 3/28: Galois Theory, Pinter Ch. 31 & Discussion of homework #7 problems
Wednesday 3/30: Galois Theory, Pinter Ch. 32 & Discussion of homework #7 problems
- Homework 7:
- Prepare to lead discussion of following problems in class Monday 3/28 - Pinter Ch. 31/ C1, C3, C9, D4
- Prepare to lead discussion of following problems in class Wednesday 3/30 - Pinter Ch. 31/ A3, A6, E4, E6, E8
- LaTeX and turn in your solutions to Pinter Ch. 31/ C1 (and two of C9, D4, E8 for extra credit) by Friday 4/1
- Other good problems to ponder: Pinter Ch. 31/A2, B3-5, C4, C5, C7, C4, E1-3,
- Research Journal Collection Friday 4/1
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| 13 |
Galois Theory |
Monday 4/4: Galois Theory, Discussion of homework #8 problems
Wednesday 4/6: Galois Theory, Pinter Ch. 32 & Discussion of homework #8 problems
- Homework 8:
- Prepare to lead discussion of following problems in class Monday 4/4 - Pinter Ch. 31/ A3, A6, (E4 & E8), (F1&F7), (G1-2), (G3-5) (parenthesis denote a single presentation)
- Prepare to lead discussion of following problems in class Wednesday 4/6 - Pinter Ch. 31/ K1, K2
- LaTeX and turn in your solutions to Pinter Ch. 31/ A3, K1, K2, due Tuesday 4/12
- Extra Credit: Prove proposition 1 (see below)
- Definition: A field F is perfect if either char(F)=0 or the Frobenius endomorphism x ↦ xp is an automorphism of F (i.e every element of F can be written as the pth power of some element in F).
- proposition 1 (a slight abstraction of F1&F7): In a perfect field, every irreducible polynomial is separable. (You should incorporate the proofs for F1 and F7, not assume them, in your proof.
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| 14 |
Galois Theory |
Monday 4/11: Galois Theory, Pinter Ch. 32, presentation of E8, K1, K2, and fixfields and fields theorem, statement of the Fundamental Theorem of Galois Theory
Wednesday 4/13: SSRD - No class
- Homework 9:
- For Monday 4/11 - Read over these notes on Galois Correspondence and prepare to guide the class in proving that fixfields are intermediate fields (aka subfields of K that contain F) - see notes for definition of fixfield. Fill out the week 14 survey: https://forms.gle/SBtZ1dSMD8bzYN5T7
- Perusall Assignment: Galois Groups & the Langlands Program, due Friday 4/15
- Good problems to ponder: Pinter Ch. 32/ set A, set B, set D, G3, G4, G5, G7, I5, I6, J1, J5
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| 15 |
Galois Theory & Group Celebration of Knowledge |
Monday 4/18: Proof of the Fundamental Theorem of Galois Theory, FTGT/ P1, P2, P. Ch. 32/ I5, I6
Wednesday 4/20: Group Celebration of Knowledge
- Homework 10:
- For Monday 4/18 - Read over the Fundamental Theorem of Galois Theory (FTGT) notes and prepare to lead discussion of following problems in class - FTGT notes/ P1, P2 and Pinter Ch. 32/ I5, I6
- Sign up as a group to find the Galois group of a given field extension and verify the Galois Correspondence: https://docs.google.com/document/d/1ZAA2Z_6RRf-S-1eQYXR2TxHUA0LjBzUxcvKzwN43Lsc/edit?usp=sharing due
Tuesday 4/19. Friday 4/22
- Study for Group Exam on Wednesday 4/20. Look over assigned and suggest problems from Pinter Ch. 31 & 32.
- LaTeX and turn in your solution to FTGT notes/ P2 by Friday 4/22.
- Other good problems to ponder: Pinter Ch. 32/ set A, set B, set D, G3, G4, G5, G7, I5, I6, J1, J5
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| 16 |
Galois' Solvability Criterion |
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