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AR01 Why Rings? Diophantine Equations & Rings
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In #abstractalgebra why do we study rings? When solving #diophantineequations we are naturally led to working with sets of numbers (quadratic integer rings) as if they were integers. This leads us to motivating questions for ring theory. In this video, three examples of Diophantine equations are considered in some detail to show that we need to define divisibility, #primenumber and prime factorization in the context of general rings. This video provides the motivation for the next videos in this series that present basics of undergraduate #ringtheory Subscribe @Shahriari for math videos at the undergraduate level.
00:00 Introduction
00:32 What is a ring?
02:58 Rings vs Groups
05:04 What is a Diophantine Equation?
05:52 Solving Diophantine Equations is hard: Fermat's Last Theorem vs Euler's Conjectures
08:24 First Example: z^3 = y^2-4 and y odd
11:26 Discussion: concepts used in 1st example
11:55 Second Example: z^3 = y^2 + 2
14:25 Definition: Quadratic integer rings
14:45 The Norm Function
15:33 Second Example continued
19:38 Nagging Questions: Primes & their properties
19:48 Third Example: z^2 = y^2 + 5
25:54 The solution to 3rd example is wrong!
26:15 What went wrong? Unique Factorization is the key
28:03 Guiding Questions for Ring Theory
A series of lectures for a full course on undergraduate abstract algebra based on my book:
Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and five time winner of Pomona College's Wig teaching award.
00:00 Introduction
00:32 What is a ring?
02:58 Rings vs Groups
05:04 What is a Diophantine Equation?
05:52 Solving Diophantine Equations is hard: Fermat's Last Theorem vs Euler's Conjectures
08:24 First Example: z^3 = y^2-4 and y odd
11:26 Discussion: concepts used in 1st example
11:55 Second Example: z^3 = y^2 + 2
14:25 Definition: Quadratic integer rings
14:45 The Norm Function
15:33 Second Example continued
19:38 Nagging Questions: Primes & their properties
19:48 Third Example: z^2 = y^2 + 5
25:54 The solution to 3rd example is wrong!
26:15 What went wrong? Unique Factorization is the key
28:03 Guiding Questions for Ring Theory
A series of lectures for a full course on undergraduate abstract algebra based on my book:
Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and five time winner of Pomona College's Wig teaching award.
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