Welcome to Math 1100!
(additions to this web site no longer count towards good deed points)
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Week of...
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Notes and Links
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| 1
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Sep 8
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About This Class; Monday - Non Commutative Gaussian Elimination; Thursday - the category of groups, automorphisms and conjugations, images and kernels.
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| 2
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Sep 15
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Monday - coset spaces, isomorphism theorems; Thursday - simple groups, Jordan-Holder decomposition series.
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| 3
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Sep 22
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Monday - alternating groups, group actions, The Simplicity of the Alternating Groups, HW1, HW 1 Solutions, Class Photo; Thursday - group actions, Orbit-Stabilizer Thm, Class Equation.
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| 4
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Sep 29
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Monday - Cauchy's Thm, Sylow 1; Thursday - Sylow 2.
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| 5
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Oct 6
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Monday - Sylow 3, semi-direct products, braids; HW2; HW 2 Solutions; Thursday - braids, groups of order 12, Braids
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| 6
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Oct 13
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No class Monday (Thanksgiving); Thursday - groups of order 12 cont'd.
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| 7
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Oct 20
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Term Test; Term Test Solutions on Monday, HW3; HW 3 Solutions; Thursday - solvable groups, rings: defn's & examples.
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| 8
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Oct 27
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Monday - functors, Cayley-Hamilton Thm, ideals, iso thm 1; Thursday - iso thms 2-4, integral domains, maximal ideals, One Theorem, Three Corollaries, Five Weeks
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| 9
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Nov 3
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Monday - prime ideals, primes & irreducibles, UFD's, Euc.Domain[math]\displaystyle{ \Rightarrow }[/math]PID, Thursday - Noetherian rings, PID[math]\displaystyle{ \Rightarrow }[/math]UFD, Euclidean Algorithm, modules: defn & examples, HW4, HW 4 Solutions
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| 10
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Nov 10
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Monday - R is a PID iff R has a D-H norm, R-modules, direct sums, every f.g. module is given by a presentation matrix, Thursday - row & column reductions plus, existence part of Thm 1 in 1t3c5w handout.
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| 11
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Nov 17
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Monday-Tuesday is UofT's Fall Break, HW5, Thursday - 1t3c5w handout cont'd, JCF Tricks & Programs handout
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| 12
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Nov 24
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Monday - JCF Tricks & Programs cont'd, tensor products, Thursday - tensor products cont'd
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| 13
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Dec 1
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End-of-Course Schedule; Monday - tensor products finale, extension/reduction of scalars, uniqueness part of Thm 1 in 1t3c5w, localization & fields of fractions; Wednesday is a "makeup Monday"!; Notes for Studying for the Final Exam Glossary of terms
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| F
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Dec 15
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The Final Exam
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| Register of Good Deeds
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Add your name / see who's in!
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See Non Commutative Gaussian Elimination
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This assignment is due at class time on Thursday, November 20, 2011.
Solve the following questions
Problem 1. (Klein's 1983 course)
- Show that the ideal [math]\displaystyle{ I=\langle 3,\, x^3-x^2+2x-1\rangle }[/math] inside the ring [math]\displaystyle{ {\mathbb Z}[x] }[/math] is not principal.
- Is [math]\displaystyle{ {\mathbb Z}[x]/I }[/math] a domain?
Problem 2. Prove that a ring [math]\displaystyle{ R }[/math] is a PID iff it is a UFD in which [math]\displaystyle{ \gcd(a,b)\in\langle a, b\rangle }[/math] for every non-zero [math]\displaystyle{ a,b\in R }[/math].
Problem 3. (Lang) Show that the ring [math]\displaystyle{ {\mathbb Z}[i]=\{a+ib\colon a,b\in{\mathbb Z}\}\subset{\mathbb C} }[/math] is a PID and hence a UFD. What are the units of that ring?
Problem 4. (Dummit and Foote) In [math]\displaystyle{ {\mathbb Z}[i] }[/math], find the greatest common divisor of [math]\displaystyle{ 85 }[/math] and [math]\displaystyle{ 1+13i }[/math], and express it as a linear combination of these two elements.
Problem 5. (Klein's 1983 course) Show that [math]\displaystyle{ {\mathbb Z}[\sqrt{10}] }[/math] is not a UFD.
Problem 6. (Hard!) Show that the quotient ring [math]\displaystyle{ {\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle }[/math] is not a UFD.
