Galois theory: Introduction

This lecture is part of an online course on Galois theory.

This is an introductory lecture, giving an informal overview of Galois theory.
We discuss some historical examples of problems that it was used to solve, such as the Abel-Ruffini theorem that degree 5 polynomials cannot in general be solved by radicals, and Wiles's proof of Fermat's last theorem.

The classic book "Galois theory" by E. Artin has been reprinted by Dover and is strongly recommended.

Correction: As pointed out by ben1996123 the product for Delta at 16:55 is missing a 24th power.

[jakelabete]

Behold a lecture by Fields medalist. What did we do right to deserve this.

[carlostrujillo]

Professor Borcherds, I took your Galois Theory class back in the early 2000s. This video is a real treat and very nostalgic. Thank you for posting this. I look forward to watching all of these over the coming months. Take care.

[rbr]

My formal and first introduction to any type of mathematics gave me the impression that it was boring and just plain uninteresting. I now believe that most people who either don't like or hate math do so only because their first encounter was bad. My teachers, unintentionally, presented mathematics as the simplest function of memorization of disjointed concepts and solving problems by following instruction (some of my teachers will not even accept a solution that was outside of their instruction) instead of logic. I wish I had more access to people like you. I am now turning 31 and has just started my journey to learning true mathematics. Keep the great contents coming!

[donaghcoffey]

Bless you. Thank you immensely for going to the trouble of doing this series.

[lawrencema]

Wow, we have Richard Borcherds giving a lecture on Galois theory. Borcherds is one of the top mathematicians in the world having won the Fields Medal (arguably the highest honors for a mathematician) in 1998.

[oskuh.]

These online lectures are absolutely great for a beginner like me. Especially the group theory series!

[henkvdpol]

enjoying the historical facts in your lectures!

[sidkulkarni]

I love these intro videos, it's like a buffet of vastly interesting topics. I look forward to deepening my understanding of Galois theory, particularly gaining an intuitive understanding of _why_ the ability to decompose the Galois group into abelian groups means that the polynomial in question is solvable in radicals.

[Viewpoint]

I just discovered your channel. It has just enough simplicity and complexity to satisfy people who have studied about 4 years of mathematics and are thirsting to really understand the interesting parts of mathematics.

[Sharpgamingvideos]

Woohoo! Just started looking into this topic these past few weeks after my fall semester first course in algebra. What great timing

[MatheusDeLimaDilima]

Hello, I'm from Brazil and I'm glad you could make it. This class is so clear, it was really a joy to watch.

Keep up the good work, cheers!

And thank you for the class.

[swaruppaul]

Thank you so much professor for starting this course. I am awestruck by this lecture. Eagerly waiting for the next lecture.!!!

[lilylikesmarkies]

Very much looking forward to the rest of this series. I wasn't aware of the connections to algebraic topology and fundamental groups, sounds like a very interesting direction to explore.

[ben]

the product at 16:55 is missing a 24th power, should be q product (1-q^n)^24

[smbcoik]

"Galois managed to die in..." I love how this is phrased

[jordimartinez]

These lectures are such a treat. I don't remember much of the mathematics I studied over 20 years ago and I am looking forward to watching the whole series. Thanks very much for posting them and sharing your knowledge with the rest of the world, this is very generous of you. I have noticed you haven't posted videos for a long time. Are you going to create more content?

[sergioestan]

Absolutely amazing introductory video. I am more inclined towards analysis, yet this video has made me want to study Galois theory and algebra in general in greater depth. Btw really cool historical notes on the side

[sachalucienmoserferreira]

Nice to find your channel on yt, its a great pleasure to be see a lecture from a field medallist. Hugs from Brazil, keep going on your videos for sheare with us a bit of your knowledgment.

[KripkeSaul]

Thank you for putting these online. I wish others would follow your example!

[JethroDjan]

this is gold! Is this how it feels to be taught my a field's medalist?