Видео

Interesting...

Hey thanks! That's a good question, I bet you could find something on StackExchange. Maybe Matlab can do something like this

if you really wanna see something, consider the definition of e as expanded by the binomial theorem with respect to the Levi-Civita field. you'll find that by definition e cannot be Real, since its definition is a non-Real Levi-Civita series. this hints at a deeper fact, which is that there are no transcendental Reals, since by definition all transcendentals incorporate infinitesimals into their magnitudes, and infinitesimals are non-Real. this then means that the Reals are necessarily a subset of the algebraics, with the remainder being Imaginary or Complex, except that's not how anyone thinks of it, because e and pi, etc. are dumbly asserted to be Real because their gross magnitude is not infinitesimal. there's nothing about v5 or i that poses any issue, those are both totally ordinary algebraics. the interesting thing is that the underlying philosophy is broken because mathematicians have no idea how to apply rigor, and instead just say random crap like 'pi is a transcendental Real number' without any evidence and then it becomes unquestionable dogma for 130 years after the tools to demonstrate that it's nonsense have been available.

@@sumdumbmick sir, this is a wendys.

Right... There is no Galois group R/Q. I guess the title means "the group of automorphism of R that fixes Q is trivial".

Yeah... I am debating whether I should reupload the video to fix that haha. "Aut(R/Q)" is probably better

Yes I think you can.

If you choose F_2 the polynomial becomes x^4+1, which is reducible because it has 1 as a root, so you do have to go slightly further to F_5. (F_3 doesn't work either)

I came here to say this.

Yep, people always seem to forget about that theorem but you can definitely use it, I think I even used it in another video haha.

not a polynomial with coefficients in Q, since that polynomial would only have finite roots.

@@penguinlord3918 yeah that makes sense

I will agree with the existing comments :)

I'm assuming by Q(i) you mean Q adjoin i, in which case the inclusion R>Q(i)>Q doesn't hold because i is a complex number? It is definitely true that R is an infinite dimensional extension of Q, but that doesn't matter for the group of automorphisms because it's not a Galois extension (which I think is what you were getting at).

@@coconutmath4928 yeah excuse my monkey brain, I was looking for any extension of degree 2 from Q in R like sqrt(2), and took the one that doesn't even land in R, my bad. If anything it confirms my initial proposition: "Galois theory has been a while for me" 😄 So yeah imagine that entire argument with sqrt(2). And the answer is, the entire thing isn't galois, so we can't use arguments of degrees of towers..?

True, here's a quick explanation of why the other direction is true. If the splitting field were properly contained in K := Q(i,\sqrt[4]{2}), then because K is a degree 8 extension the splitting field would have degree less than 8 and therefore the Galois group would have size smaller than 8. By computing the Galois automorphisms directly though, you can see there are exactly 8 of them, because there are four ways to map \sqrt[4]{2} and two ways to map i to something. When I took Galois theory we started by doing splitting fields then talked about Galois groups, so I wanted to keep those topics separate, but it would probably have been more rigorous to talk about the Galois group here as well.

Modular forms is a great idea. I will get some videos uploaded soon. Algebraic topology is something I might learn more about next year, in which case I will give that a shot on my channel :O

You can learn more about that here if you're interested: ruclips.net/video/HX-IfYxOui0/видео.html&lc=Ugyinyg1K4n2eaSiyyB4AaABAg&ab_channel=CoconutMath You can apply everything in that video to this problem by thinking about subgroups of G.

Thanks for the support! Yeah I mostly learned Galois theory stuff by just doing examples... then there are the theorems about algebraic closures etc. that are a lot harder to understand haha. Glad you're enjoying!

That means a lot!! I will be back with more videos on this soon. I always appreciate suggestions for problems if you have any

Thanks very much for the support! In the near future the plan is modular forms (not sure exactly what yet) and more Galois theory problems (of which two were just uploaded).

OK :)

I use microsoft whiteboard and record with OBS studio. OBS is free and records via window capture. Whiteboard is not great but gets the job done haha

Thanks bro 👍 You are doing great job

Thanks very much!

Thanks :)

Good idea, let me put some of the Galois theory vids together. Thanks for the support

Thanks very much! I think one reason is because the topics are niche. Only so many people who want to learn number theory/galois theory/etc. Hoping to get more in the future though

I can try, what is your question?

Thanks! Not sure if you found it already but there's a video about that on my channel.