What is Solvability in Galois Theory?

[@geneeditor9545]

I loved Sofia playing a more active role in the video. Since math has a long and interesting history her background may help a lot too. You both are doing a great work !

[@dibeos]

@@geneeditor9545 Hey, Sofia here. Thanks so much for your comment! I scripted the video, so I’m glad it could help 😉

[@TheFirstTheLast-g4t]

This is the most promising video i have seen about this subject, thank you very much for that! Still something is missing. i wish there were more examples a long the way. They help. In their absence, i fail to grasp several milestone. For instance when you talk about permutation of the roots, i don't get the significance because i don't see where/ how the permutations are used, so why are we looking at those permutations ? or when you say the big group should consist of solvable subgroup, i don't know how subgroups are defined etc. I strongly wish there will be a video that help a layman like me really understand. All the best to the couple!

[@dibeos]

@@TheFirstTheLast-g4t thanks for the comment! Really!! I was thinking about what you are saying… also based on other comments in the channel. But now that I read this comment I think I understand better the problem. Please give us your advice: do you think people would grasp the concept better if we made a video ONLY showing (like ~20) examples of solvable and unsolvable groups? If not please give us some good idea, we really want to help people with that

[@fernandojackson7207]

A subgroup S of a group G is a subset of G which is itself a group. One result to help determine whether S is a subgroup is if s1,s2 are in S, then s1*(s2)^{-1} is also in S. Here * is the group operation , and (S2)^{-1} is the inverse of s2.

[@DanHorus899]

I was waiting to see this video for so long, glad you explained it.

[@dibeos]

@@DanHorus899 awesome!! Is it clear now? Would you like a video on another topic as well? 😎

[@burkhardstackelberg1203]

I indeed would love to learn the fully-fledged Galois theory one day... As a physicist with a knack to deep mathematics, I would expect a connection to the classification of simple finite groups, as it would give tools for solutions with higher functions, i.e. outside the toolkit of elementary functions/radicals.

[@thedude882]

Thank you for covering Galois theory, I find it one of the most beatiful areas of mathematics.

[@dibeos]

@@thedude882 we noticed that many people (just like Sofia and I) are very interested on Galois Theory, so we decided to post more often about it from now on 😎

[@Minu-yt2xw]

Don't stop making videos this is so informative and it develops my intrest towards math ❤!

[@dibeos]

@@Minu-yt2xw thanks Ella!!! It means a lot to us!!! 😎💪

[@hambonesmithsonian8085]

Yes please talk about field extensions. (Maybe even mention how they’re also vector spaces!)

[@dibeos]

@@hambonesmithsonian8085 yes, we will do it 😎

[@jujoropo]

Epic stuff!!! 🎉

[@fbachan]

At 7:15 you say "Group G is solvable if there exists a series o subgroups ..." This is the point where I got lost, you explain too quickly and without giving an example. I try to re-play more times, thank you!

[@dibeos]

@fbachan thanks for the feedback! Other people told us the same thing. We are trying to explain things slower and using more examples now. Let us know if in our most recent videos we actually fixed that 😆

[@user-wr4yl7tx3w]

great content! really amazing the connection between polynomial and groups

[@dibeos]

@@user-wr4yl7tx3w we are glad you liked! Thanks for the nice comment 😎

[@AltAaltonnov]

I would like to see an example where the galois theory is used as I think it's meant to be used: to find the Galois group without knowing the roots. If you can easily solve the equation, you already know that the Galois will be solvable. I also would like to see an example where the Galois group of a nontrivial fifth degree polynomial is found to be solvable at least pretending not knowing the solution, and then this information is used to solve the equation. This might not be "easy," but I remember seeing such an example a long time ago...

[@Satisfiyingvideo-uu9pw]

Thank you sir. Can you give a video about laplace transfrom.

[@dibeos]

@@Satisfiyingvideo-uu9pw yessss!!! I’m gonna add to our list right now!! 😎

[@aaronmartens2903]

This video offers an overview into Galois Theory. I would personally appreciate examples to build intuition about the symmetries of a polynomial.

[@dibeos]

@@aaronmartens2903 hi! Yes, many people asked for concrete examples in other comments in the channel. We will probably make a 10 minutes video ONLY on examples in Galois theory. What do you think?

[@shaneri]

Thank you Luca and Sophia! I'd like to see some math logic as well!

[@dibeos]

@@shaneri Hi Sergey!!! Yes, we will have some math logic in the channel 😎

[@SobTim-eu3xu]

Great video!) I love it

[@Satisfiyingvideo-uu9pw]

Please field extension

[@dibeos]

@@Satisfiyingvideo-uu9pw yes Sir!!! Actually I already have a raw draft of this one, so this video will come sooner than the one on Laplace Transform

[@user-wr4yl7tx3w]

are you guys like both mathematicians? how did this youtube channel get started?

[@dibeos]

@@user-wr4yl7tx3w great question. I (Luca) am a mathematician and theoretical physicist. My wife Sofia is a historian but is very interested in math and physics as well. We decided to start this channel to teach math and physics through story telling methods. I hope you like it 😎

[@Leo-if5tn]

I hope some day I will be able to study Galois Theory further!

[@dibeos]

@Leo-if5tn we want to make deeper videos on Galois Theory on the channel. The idea is to simplify the concepts as much as humanly possible 😎

[@willnewman9783]

At 3:35, you male the same weird definition you made in the last video on this topic, where you say that Sk only allow permutations of the roots which "leave the relationships of the polynomial unchanged." It is not clear what this means, and I don't think you want to add whayever you think this requirement means. Especially because at 4:26, you say Sk contains ALL permutations. 4:44 You do not explain what it means for two tuples of roots to be conjugate to each other. And I wouldn't be surprised if it is impossible to explain what this means without field theory, which you are claiming to be avoiding. 8:58 The Galois group of that polynomial is not D4, it is C2. I have no idea where you got D4 from.

[@noahtaul]

You forgot 6:32, absolutely being normal inside D8. I really want to like these videos, being an algebraic number theorist, but they just make so many mistakes that it’s clear they don’t know what they’re talking about.

[@em_zon2643]

Interesting!

[@dibeos]

@@em_zon2643 we’re glad you liked it! Let us know what kind of content you’d like to see in the channel, please 😎

[@plranisch9509]

☀️

[@sumdumbmick]

I've still never seen anyone actually explain Galois Theory. you describe it well enough to start using it, but I'm highly skeptical that you have the slightest clue why it works given how roundabout all of your attempted explanations are.

[@sumdumbmick]

note that a definition which allows something to be used is not an explanation. I understand that in modern Mathematical logic this is what everyone considers to be an explanation, but it simply fucking is not.

[@theultimatereductionist7592]

To this day, in spite of having earned my Math PhD in 2000 specializing in differential algebra, I do not understand the necessity of G(i+1) being normal in G(i) to the fields fixed by these groups, nor the necessity of the quotient group G(i)/G(i+1) being abelian. What happens if G(i+1) is normal in G(i), so but G(i)/G(i+1) is a non-abelian group? What goes wrong with the solvability by radicals?

[@dibeos]

@@theultimatereductionist7592 To answer this question, one needs to understand the necessity of certain conditions in Galois Theory related to solvability by radicals. Some of them: (1) Necessity of G(i+1) being normal in G(i): In Galois Theory, the normality condition ensures that the extension fields corresponding to these groups form a tower of fields with specific properties that can be analyzed through their automorphisms. If G(i+1) is not normal in G(i), then the intermediate field extensions won't align properly with the group structure, disrupting the analysis of their automorphisms. (2) Necessity of G(i)/G(i+1) being abelian: The quotient group G(i)/G(i+1) being abelian is crucial for the process of radical solvability. If each successive quotient in the derived series is abelian, it allows the use of radical extensions, which are necessary for expressing the roots of polynomials in terms of radicals. Non-abelian quotients disrupt this structure and prevent the expression of roots through radicals. (3) Implications of G(i)/G(i+1) being non-abelian: If G(i+1) is normal in G(i) but G(i)/G(i+1) is non-abelian, the resulting group structure doesn't support solvability by radicals. The radical expressions rely on the abelian property to simplify the extensions and solve the polynomial equations step by step. Non-abelian groups introduce complexities that cannot be resolved within the framework of radical extensions. In other words, both the normality of G(i+1) in G(i) and the abelian nature of G(i)/G(i+1) are essential to maintain the structural integrity required for solving polynomials by radicals. Deviating from these conditions disrupts the hierarchical approach necessary for such solutions. Let me know if I understood your question correctly and if I answered it. Of course, in order to be more detailed I would need to delve deeper into field extensions.