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/sci/ - Science & Math

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1181858 No.1181858 [Reply] [Original]

Consider the group GL(2,Q) of invertible 2x2 matrices over Q.
Does the polynomial <span class="math">x^5 - 1[/spoiler] have non-trivial (<span class="math">x \neq E_n[/spoiler]) nulls in GL(2,Q) for <span class="math">n \geq 5[/spoiler]?

I don't really think so.
For one, the orthogonal matrices could be considered:

[cos(x) -sin(x)]
[sin(x) cos(x)]

for <span class="math">x = \frac{2\pi i}{n}[/spoiler]. But I'm quite sure that
(cos(x), sin(x)) <span class="math"> \notin Q^2[/spoiler] given the above definition of x for <span class="math">n \geq 5[/spoiler].

Are there other possibilities?

>> No.1181861

>>1181858
Of course. I agree with you

>> No.1181863

>>1181858
*actually, consider the ring of 2x2 matrices over Q.
Is there then any non-trivial invertible solution to <span class="math">x^5 - 1[/spoiler]?

>> No.1181865

>>1181863
Aaah, yes of course! Blast.. why didnt I think of that! Oh well.. that is why we are here isnt it! to LEARN!

>> No.1181867

*lulz confusion. I wrote <span class="math">x^5 - 1[/spoiler] when I meant <span class="math">x^n - 1[/spoiler] for <span class="math">n \geq 5[/spoiler].

>> No.1181874

I also wrote
"non-trivial (<span class="math">x \neq E_n[/spoiler])"
when I meant
"non-trivial (<span class="math">x \neq E_2[/spoiler])"

man, I should go back to bed.

>> No.1181875

>>1181867
good, I was testing you!.. seeing how long it would take you to pick up on your slip up there!

>> No.1181880

>>1181874

Tiring gets us all.. dont worry.. you just need to follow this table I'll draw up for you

X= Sleep
Y = Brain functionality

- Y + X*Multiplied by* 8hrs = ++Y

Its brilliant! OH HOW I LOVE TO LEARN

>> No.1181889

Is your <span class="math">E_2[/spoiler] the identity matrix?

>> No.1181891

>>1181880
>>1181875
>>1181865
>>1181861
notice how I haven't responded to you yet?
Can you figure out why that is?

>> No.1181893

>>1181889
Yes. The question is: is there a matrix in GL(2,Q) which risen to the n-th will yield the identity matrix for <span class="math">n \geq 5[/spoiler].

I really suck at formulating stuff.

>> No.1181897

>>1181891
Notice how I couldnt care less.. I'm sleep deprived and i cant be fucked putting up with the buzzing on /b/ and /r9k/, also /x/ and /lit/ are slow tonight. So I'm adding to the brain power of my fellow /sci/entific chaps.

>> No.1181898

>>1181893
Aye, I understood. I don't know a lot about rational matrix multiplication. But maybe you can use diagonalization? It's clear that for n uneven the determinant has to be 1. And for even n it can also be -1.

>> No.1181902

i'm 12 and what is this?

>> No.1181901

>>1181893
I want to find an <span class="math">A \in [/spoiler] GL(2, Q) with
<span class="math">A^n = E_n[/spoiler] for an <span class="math">n \geq 5[/spoiler].
An element of GL(2,Q) with order 5 so to speak.

And that's the last time I'll restate my question. I'm sure it's clear what I'm talking about now.

>> No.1181899

>>1181893
You tell that Idiot.. now.. the reason we're here.. SCIENCE! Continue, friend..

>> No.1181907

>>1181902
This.. is SCIENCE! Lucky you came here at a young age.. Keep under our wing and we can guide you!

>> No.1181909

>>1181858
protip: you know n roots. there are no other roots.

you need to prove that those n are irrational

>> No.1181913

>>1181909
DARN IT STRAIGHT TO HECK! I was just about to suggest this. oh lolololsciencelololol

>> No.1181916

>>1181913
Um, can you please just leave.. you're not accomplishing anything

>> No.1181921

>>1181916
its alright friend, this may be a hard problem to solve, but just because its difficult doesnt mean we're accomplishing nothing, our efforts are not futile.. SCIENCE IS ALL ABOUT TRIAL AND ERROR!

>> No.1181933

>>1181909
You mean the powers of the orthogonal matrices?
I think you're right. It's pretty analogous to primitive roots of unity.

So I have to show that <span class="math">(\sin \frac{2k\pi}{n}, \cos \frac{2k\pi}{n}) \notin Q[/spoiler] for all <span class="math">1 \leq k < n[/spoiler] and <span class="math">n \geq 5[/spoiler].

I think I'm just gonna believe that :d

>> No.1181938

>>1181933
*<span class="math">\notin Q^2[/spoiler] for fucks sake.
today fucking fucks.

>> No.1181945

>>1181938
fuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckSCIE
NCEfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckf
uckSCIENCEfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfu
ckfuckfuckSCIENCEfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuckfuc
kfuckfuckfuckfuckSCIENCEfuck

>> No.1181949

Dude nobody on /sci/ can help you with that nobody here knows anything of trigonometry.

>> No.1181951

>>1181949
>>Science & Math

>> No.1181974
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1181974

>>1181933
it is pretty simple to show that they are irrational

>> No.1181990
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1181990

Since nobody cares, cereal guy disappears

>> No.1182317
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1182317

get back in here, this instant.