Abstract Algebra, 30B: Integral Domains: Irreducibles, Primes, PIDs, UFDs, Fermat's Last Thm, EDs

(0:00) Proof that 1+sqrt(-3) is irreducible but not prime in Z[sqrt(-3)].
(4:49) Theorems relating irreducible elements and prime elements in integral domains and principal ideal domains.
(6:51) Unique factorization domains.
(8:55) Every PID is a UFD (so F[x] is a UFD, where F is a field).
(10:19) Z[x] is a UFD even though it is not a PID.
(12:30) Z[sqrt(-5)] is not even a UFD (for example, 21 does not factor uniquely as a product of irreducibles).
(14:01) History of Fermat's Last Theorem leading up to Andrew Wiles' proof (also note contributions from Jean Luc Picard and Homer Simpson...lol ).
(21:14) Brief overview of Euclidean domains (including the facts that every ED is a PID and therefore every ED is also a UFD; if D is a UFD, then D[x] is a UFD (like for D = Z); the Gaussian integers Z[i] form an ED).
(23:00) Mathematica code to visualize Z[sqrt(d)], when d is negative, and the level curves (contour map) of the norm N (with formula N(a + b*sqrt(d)) =a^2 + d*b^2).

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[niazazeez]

This is clearly the best video series on Abstract Algebra online. Unfortunately I discovered it quite late, at the of Week 5 of my 8 week accelerated course (and this was my first AA course so it was hard). But I intend to take more AA courses, and intend to go back and watch everything from the beginning, and whatever I can get done before my final will be a great review.

[sherryj]

At 25:27, norm of 1+sqrt(-3 ) should be 4 not 10 .Please see.

[sherryj]

Hi Sir, At 1:48 we have concluded N(x)=N(y)=2.Then, you have assumed x, y as both equal and there value be a +bsqrt(-3).But, if norms of two elements are equal, it does not always hold that elements are equal .eg, Norm of 2 and - 2 will be 4 but 2 is not equal to -2 .So, why have you assumed x, y as equal .Please clarify.
Thanks