Course Overview
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Course Synopsis
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This course is a comprehensive introduction to commutative algebra. It is meant to give students a foundation for further studies in algebra, combinatorics and algebraic geometry. You will understand algebra can be applied to other important areas of sciences.
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Course Learning Outcomes
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At the end of the course, students will be able to:
- Gain familiarity with the polynomial ring and be able to perform basic operations with both elements and ideals.
- Use computational tools, especially Gr?bner bases and the Buchberger algorithm, to solve problems in polynomial rings; for example the ideal membership problem, or finding solutions to polynomial equations.
- State accurately and be able to explain the proofs of the main results in the class without access to notes or other resources.
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Course Calendar
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6
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Identity/inverse uniqueness
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18
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Operations on polynomial
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19
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K-Algebra homomorphism
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20
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K-Algebra homomorphism example
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21
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Homogeneous polynomials
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23
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Proposition (graded ideal)
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24
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Polynomial ring result
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28
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Monomial ideals reasult-1
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29
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Monomial ideals reasult-2
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31
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Dickson's lemma results-1
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32
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Dickson's lemma results-2
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33
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Operations on monomial ideals
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34
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Power of monomial ideals
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35
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Intersections of monomial ideals
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36
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Intersections of monomial examples
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37
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Quotient of Monomial ideals
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38
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Quotient of ideals examples-1
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39
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Quotient of ideals examples-2
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40
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Square free monomial ideals
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41
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Radical ideal criteria
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42
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Radical of a monomial ideal
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43
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Irreducible monomial ideals
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44
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Irreducible monomial ideals 2
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45
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Irreducible monomial ideals example
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46
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Irreducible monomial ideals example 2
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47
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Irreducible monomial ideals example 3
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50
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Primary decomposition example 1
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51
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Primary decomposition example 2
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52
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Primary decomposition example 3
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53
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Introduction to grobnor bases
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55
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Algebraic variety of an ideal
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56
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Ideal membership problem
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58
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Hilbert bases theorem-1
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59
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Hilbert bases theorem-2
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60
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Hilbert bases theorem-3
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63
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The one variable case 1
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64
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The one variable case 2
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65
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The one variable case 3
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66
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The One Variable Case (E.A)
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67
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The One Variable Case (E.A) 2
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68
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The One Variable Case 5
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70
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Term Orders of Proposition
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72
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The Degree Lexicographical Order
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73
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The Degree Reverse Lexicographical Order
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74
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Term Orders Proposition 2
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83
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Ideal Membership Problem Solution
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85
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Buchberger's algorithm-1
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86
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Buchberger's algorithm-2
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87
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Grobner bases example-1
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88
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Grobner bases example-2
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89
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Minimal grobner bases-1
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90
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Minimal grobner bases-2
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93
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Applications of Grobner Bases-1
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94
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Applications of Grobner Bases-2
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95
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Applications of Grobner Bases-3
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96
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Applications of Grobner Bases-4
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97
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Applications of Grobner Bases-5
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98
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Applications of Grobner Bases-6
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99
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Applications of Grobner Bases-7
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100
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Simplicial complexes-01
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101
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simplicial complexes-02
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102
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Examples simplicial complexes
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103
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Simplicial complexes example 1
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104
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Simplicial complexes example 2
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105
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f- vector of simpicial complexes
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106
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f- vector of simpicial complexes 2
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107
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f- vector of simpicial complexes 3
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108
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Kruskal Katona theorem
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109
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Kruskal Katona theorem P1
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110
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Reverse lexicographical ordering
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111
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Kruskal Katona theorem P2
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113
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Stanley Reisner ring 1
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114
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Stanley Reisner ring 2
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120
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Shellable Simplicial Complex
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