Visual Group Theory: Lecture 7.5: Euclidean domains and algebraic integers

Visual Group Theory: Lecture 7.5: Euclidean domains and algebraic integers.

Around 300 BC, the Greek mathematician Euclid found an algorithm to compute the greatest common divisor (gcd) of two numbers. Loosely speaking, a Euclidean domain is a commutative ring for which this algorihm still works. It is not difficult to show that these rings must be PIDs. One common example are the Gaussian integers, which are the complex numbers with integer coefficients. A generalization of this are the algebraic integers, which is the subring algebraic numbers that are roots of monic polynomials in Z[x]. We are particularly interested in those that are contained in quadratic extension fields of the rationals, Q. In this lecture, we analyze these rings, and try to understand which are Euclidean and which are PIDs. Both of these complete classifications are still open, but we review what is known. We conclude with several pretty pictures of algebraic integers in the complex plane, and a updated diagram showing the relationships between the types of rings we've seen so far: fields, Euclidean domains, PIDs, UFD, integral domains, and commutative rings, and how they fit together.

[zachdingman]

For whatever reason groups have been like brain Velcro for me for the last year or so, I can’t understand them very well but I also can’t stop thinking about them, so thank you for creating a form of entertainment that helps to bridge the gap!

[martinkrajci]

Thank you Professor for this course. Definitely the best one on youtube on the topic. Finally all the pieces clicked together. Nice work

[joymenezes]

Thanks Professor Macauley.
I had really hard time understanding concepts of abstract algebra. But due to this lecture series I've understood most of it.

[asdfghqwerty]

thanks professor. I see full playlist in 1 week before restart my algebra course at university . Now I understand a lot of concepts that at university were dark for me. grazie mille ( 1000 thanks, in Italian )

[sleepymalc_deprecated]

I had never studied abstract algebra before (this keeps bothering me for my journey of mathematics), hence I decide to run through this in 4 days. Finally, it comes to an end! This is especially helpful for me since I have lots of pieces of knowledge about algebra when studying discrete mathematics and other topics, nice lectures overall for me to wrap them up. The only part I'm not that satisfied with is that we skip the proof(not even outlined) of the Galois Theorem, which I think it's the biggest motivation for me to study algebra, but the related material can be easily found online, so not a big issue!

[koenigmagnus]

12:04 should be x*f(x, y) + y *g(x, y)

[GiovannaIwishyou]

Well, this was the longest how to solve a Rubik's cube manual... 🤔

[Jkfgjfgjfkjg]

@1:12 "...where r is less than a" should be "...where r is less than b".

[griddlebone]

could you please share also 7.6 and 7.7?

[partialorder]

@26:40, shouldn't it be 11^7 +11^6+11^5+...+11 polynomials (11 choices for each of the 7 non-leading coefficients of 7th degree polynomials, plus 11^6 choices for the 6th degree polynomials, etc)? This is just over 20 million - which is much more feasible to work with than 7^11 (just under 2 billion).

[hyperduality]

Automorphism = Duality -- conjugate root theorem.
"Sith lords come in pairs (duals)" -- Obi Wan Kenobi.
Injective is dual to surjective synthesizes bijective or isomorphism.
Subfields are dual to subgroups -- the Galois correspondence.
"Always two there are" -- Yoda.
Brahman (thesis, the creator God) is dual to Shiva (anti-thesis, the destroyer God) synthesizes Vishnu (the preserver God) -- the time independent Hegelian dialectic.
Hinduism is consistent with the Hegelian dialectic!
Duality (thesis, anti-thesis) creates reality (non duality).