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Really glad to see the small finite sets without metrics make it to the examples. Textbooks and professors rarely address them and expect students to divine on their own. I wish you had an example of a non-bijective function and how continuity may fail.

The metric space criteria for convergence are clear. The general topological space criteria for convergence are not. It would have been more helpful to use as example the usually featured set X = {1, 2, 3} and its usually featured topology {emptyset, X, {1}, {1, 2}) and talk about point x = 3 and how basically every sequence converges to it (and possibly another limit as well). Do I understand this correctly? The only neighborhood of 3 under this topology is the set X. So no matter what the sequence is, X is always a neighborhood of any element in X, and 3 is thus always in the neighborhood. A sequence <x_n> = (1,2,1,2,1,2,1,2,1,2...) converges to 2 and to 3. Or did I misunderstand definitions? ----- Never mind, I should have watched the video till the end. Great video.

Very good video, and I appreciate the examples. Could you please order all your topology videos in a way they would be presented as lectures in a course? Thank you greatly.

FYI, because sagemath is based on Python, I believe you can use f-string formatting to print both the expression and result of an expression more easily using the equal sign, i.e. `f"{expr=}"`. For example, at 8:15, you could do `print(f"{gamma^(power)=}")`.

Amazing! Thank you

Thanks for great presentation

very nice, lets see Paul Allens Field extension

awesome video now do nxm

It is crazy

you are so under-rated!!! Your content is a gold mine for number theory in math olympiads!! You've got a new subscriber!! :)

Very cool video! I had never seen these concepts before. Also never seen Sage. In the past I would use an ancient piece of software called Maxima to do algebra, whenever I got bored of doing it by hand. It seems Sage is a newer Python-based alternative, but it looks like they haven't figured out graphical input and output of mathematical formulas yet? One question: is (³√2 + ³√3) just one example of a primitive element for the field ℚ(³√2, ³√3)? I suppose any number of the form (a ³√2 + b ³√3 + c) with a, b, c ∈ ℚ would be a suitable alternative. But do all of its possible primitive elements have this form? Also, what happens when you keep adding all possible roots to ℚ? Since both the root arguments (ℕ) and the root indices (ℕ) are countably infinite, all possible roots are also countably infinite, by a similar diagonal construction as for ℚ. (Does the set of all possible roots have a name?) So the resulting extension would be isomorphic to a countably-infinite vector space of ℚ... which sounds a lot like ℝ. Is this yet another construction or definition of ℝ? Do transcendental numbers naturally appear in it, even though they were not included in its construction? Also, the primitive element theorem clearly does not apply, because it's not a finite extension, but what would a degenerate primitive element of this infinite extension look like? The... sum of all possible roots?

Please keep clickbaiting math! 🎉

Do complex integrals and the residue theorem please

This is awesome, especially the Sage Math demo. This is the type of thing that should have been given as hw in my abstract algebra II class where Sage Math could be used instead of keeping the problems computationally simple enough to do on paper, which forces the set of available hw problems to be quite boring.

Great 👍

you should have made your first basis vector gamma^0 = 1 and not used gamma^9, but the linear combinations that make alpha and beta didn't end up using the gamma^9, so I don't have that much of an issue.

Seems like an interesting video, but so quiet that it's actually hard to watch lol

The short form content is great, enough to intrigue and a great sell to watch the full video. Gonna have to check it out, please keep sharing!

You successfully tricked me into reviewing my abstract algebra knowledge

Amazing video! What software did you use to animate it?

Title should’ve ended with ellipsis

Phrasing an abstract algebra explainer as if it's youtube drama is quite possibly the greatest thing I've ever seen. You could teach so much to unsuspecting high-schoolers by making titles like this hahaha

no need just see the equation , it say it all.

En los últimos días, he estado ocupado tratando de construir la raíz cúbica de tres en el plano. Claro, demostré que es imposible. Y luego, aparece este video recomendado, fruto de cómo me espía el algoritmo

How is it possible that this channel does not have 250 000 subscribers?

Elementary, my dear Watson!

Great video, and the history at tme beginning helped to explain the motivation for studying these field extensions. (I've only taken a basic Abstract Algebra class that never reached the topic of field extensions.) And, yes, like everyone else is saying, I clicked for the silly title too, haha!

Pretty random but I watched your Topology playlist for a whole semester, did so good in my class that my professor offered me an undergrad research opportunity :) I think I'll owe you my degree when I graduate lol

This Is The Greatest Mathematics Of All Time

Thank you kind sir! Never heard of this theorem, nevertheless loved everything about this video: historical bit, explanation, visualisations ❤

three completely horrible video titles today. it's really wearing me down

Great video, I took an abstract algebra class as a comp sci major, and loved learning more about this type of thing. Keep it up!

moistcritical physics

I thought the title meant too many primitive elements are being discovered or sth🤣

Oh my god you mentioned Brahmagupta! Finally professors have stopped teaching that the Greeks and Persians came up with everything! Mathematics has had such a rich history and india was a big part of it, its so sad that this history has been almost totally omitted up until relatively recently. Lots of love, Gaurahari Das

very deep result. Galois couldn't demonstrate it.

Here's a fun problem. Let a,b,c be formal variables. Let L be the field Q(a,b,c), and let K be the field Q(a+b+c,ab+bc+ca,abc). Clearly K is a subset of L. (In fact it is precisely the set of _symmetric_ rational functions in a,b,c.) The problem is to prove that L=K(a-b). Note that this is an instance of the primitive root theorem, in the sense that the theorem guarantees us that L=K(gamma) for _some_ gamma in L.

You are talking too fast. I can't follow at that speed.

Thanks! You math Guys are great!

2:30 It is del Ferro, not 'Farro'

the proof is what's crazy (imo), because it turns out almost all gamma = alpha+x*beta can do the job. But I don't remember if there is some clever choice of x that will always work.

this is a fun video idea! edit: im also a big fan of judson's free algebra book!

the clickbait worked 😂

Very nice video!

Super cool video! Seeing (gamma = alpha + beta) at the end was a bit jarring though: I think a common reaction to this result would inspire the question "is that just always true?". A quick statement like "in this case, it happens to be..." would help contextualize that answer since the result is presented almost as trivial without explanation. @MasterHigure has a nice comment below that added some important information to this point, though I cannot understand the notation 🙂

I think in Zorn's Lemma we need to specify that there should be an upper bound IN THE SET S, so that the statement holds. Otherwise, we can take the set [0, 1), which doesn't have a maximal element

You are such a nerd.

Liked for AD

commenting for the algorithm gods

very good demonstration 👍🏻 really looking forward to more videos from you.