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===Solve the following questions=== |
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===Solve the following questions=== |
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'''Problem 1.''' (Klein's 1983 course) |
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'''Problem 1.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>. |
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# Show that the ideal <math>I=\langle 3,\, x^3-x^2+2x-1\rangle</math> inside the ring <math>{\mathbb Z}[x]</math> is not principal. |
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# Is <math>{\mathbb Z}[x]/I</math> a domain? |
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'''Problem 2.''' (Lang) Show that the ring <math>{\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? |
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'''Problem 2.''' Prove that a ring <math>R</math> is a PID iff it is a UFD in which <math>\gcd(a,b)\in\langle a, b\rangle</math> for every non-zero <math>a,b\in R</math>. |
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'''Problem 3.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements. |
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'''Problem 3.''' (Lang) Show that the ring <math>{\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? |
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'''Problem 4.''' (Hard!) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD. |
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'''Problem 4.''' (Dummit and Foote) In <math>{\mathbb Z}[i]</math>, find the greatest common divisor of <math>85</math> and <math>1+13i</math>, and express it as a linear combination of these two elements. |
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'''Problem 5.''' (Klein's 1983 course) Show that <math>{\mathbb Z}[\sqrt{10}]</math> is not a UFD. |
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'''Problem 6.''' (Hard!) Show that the quotient ring <math>{\mathbb Q}[x,y]/\langle x^2+y^2-1\rangle</math> is not a UFD. |
Revision as of 09:51, 6 November 2014
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 PID, Thursday - Noetherian rings, PIDUFD, 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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In Preparation
The information below is preliminary and cannot be trusted! (v)
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
inside the ring ![{\displaystyle {\mathbb {Z} }[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e080fddb4f8b99adc91d4b678074184a78b306f1)
is not principal.
- Is
![{\displaystyle {\mathbb {Z} }[x]/I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0d523ee9cb85ab2b3ef004cd039aedde51e280)
a domain?
Problem 2. Prove that a ring 
is a PID iff it is a UFD in which 
for every non-zero 
.
Problem 3. (Lang) Show that the ring ![{\displaystyle {\mathbb {Z} }[i]=\{a+ib\colon a,b\in {\mathbb {Z} }\}\subset {\mathbb {C} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf6c02172f2bb115542d48db5c7d622beffc54b)
is a PID and hence a UFD. What are the units of that ring?
Problem 4. (Dummit and Foote) In ![{\displaystyle {\mathbb {Z} }[i]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2409dcaf0c37bf7184798dfd10f4a2da70d980ec)
, find the greatest common divisor of 
and 
, and express it as a linear combination of these two elements.
Problem 5. (Klein's 1983 course) Show that ![{\displaystyle {\mathbb {Z} }[{\sqrt {10}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4905152d1bea6d8056ac5d833d387feac8fea859)
is not a UFD.
Problem 6. (Hard!) Show that the quotient ring ![{\displaystyle {\mathbb {Q} }[x,y]/\langle x^{2}+y^{2}-1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d481fae3a288f265081fd3b5d7a904520ba9a71f)
is not a UFD.
