Number theory - 2014-10-30

Alizter

15:01

pwoooff

I have arrived.

Balarka Sen

@Alizter Did you see the geometric not-so-rigorous analog of galois theory above ?

Alizter

@BalarkaSen whare?

Balarka Sen

just above

Alizter

You wrote?

Balarka Sen

from the place where the snapshots begin.

from here

Alizter

15:06

@BalarkaSen Do you have the geogebra link?

I want to play with it.

Balarka Sen

web.geogebra.org/app

try it

the best way to understand is to do teh experiments yourself and observe.

Alizter

How do I plot stuff :P

Balarka Sen

ok, what do you see in the window?

Alizter

Create your own

Balarka Sen

click on algebra

Alizter

15:09

ok

Balarka Sen

now click on points and place a point.

Alizter

yup

Balarka Sen

place another point and right click on it and click on properties

Alizter

yes

Balarka Sen

what do you see?

Alizter

15:10

Basic, Color, Style etc.

Balarka Sen

do you see anything like "define"?

or "definition"?

Alizter

yes

Balarka Sen

ok. is it a textbox?

Alizter

yup

Balarka Sen

then click on the textbox, remove all that is in there and replace by "sqrt(A)"

Alizter

15:11

invalid input

Balarka Sen

your original point was named "A" right?

Alizter

yes

Balarka Sen

ok try clicking on A then and properties

click on "Algebra" tab

and change cartesian to complex

Alizter

ok

Balarka Sen

now try changin the definition of B

to just "sqrt(A)"

Alizter

15:16

working now

Balarka Sen

ok, good

@Alizter now shift A from the first coordinate to the 3rd

i mean, from the upper left to lower left

Alizter

yes

Balarka Sen

the curve traced by B is discontinuous, right?

Alizter

yup

Balarka Sen

now place another point and define that to be -B

move A similarly : B and C will switch places

Alizter

15:18

nice

Balarka Sen

that rather looks like a galois automorphism acting doesn't it?

Alizter

Or more like a branch?

Balarka Sen

yes, but that is teh complex analytic way of saying it.

look at it as a galois automorphism

it is your classical galois theory

@Alizter now start a new algebra sheet. or rather delete your B and C.

leave A alone.

Alizter

I changed it to cube root

Balarka Sen

ok, try it

you'll get similar.

Alizter

15:22

how can I get all the roots of unity to show up

it was easy for sqrt because of -

Balarka Sen

-1/2 +/- sqrt(-3)/2

Alizter

ok

Balarka Sen

@Alizter I'd recommend you to do something else. Try drawing a point A and B = A^2 on the complex plane. note that you need to make B loop twice around the origin for A^2 to come back to where it started.

you can try similar for B = A^3.

i.e., whilst A loops once, B loops twice.

however, this works only if A loops around some point on the negative real axis.

Alizter

ach why is it so slow

hmm yes

Balarka Sen

it works faster if you cancel out the properties tab

Alizter

15:30

That is silly

Balarka Sen

@Alizter what is?

Alizter

That the properties tab is causing slowness

Balarka Sen

no it is not

Alizter

@BalarkaSen I feel like I am playing with keplers laws

Balarka Sen

the props tab shows you the coordinates

that takes time

@Alizter hah

well play with B = log(A)

that is more funny

Alizter

15:32

that branch

Balarka Sen

mmhmm

the usual log is defined by abs arg log < pi

if there hadn't been that restriction, you could have seen B going above and above and notcoming back

Alizter

@BalarkaSen I like to think of log as a drill bit

Balarka Sen

:P

well, back to buisness

think of B = A^2

Alizter

B=1/A is also interesting

look how chaotic it is near 0

Balarka Sen

mildly.

Alizter

15:34

but does nothing every where else

Balarka Sen

it's the riemann sphere, dude

Alizter

I knwo.

Balarka Sen

ok, where was i?

right B = A^2

A loops once, then B loops twice

Alizter

@BalarkaSen on the imaginary axis

Balarka Sen

correct.

well, in prticular on the origin

Alizter

15:36

B=A^3 A loops once then B loops thrice?

Balarka Sen

yes.

and this reminds a bit about cyclic grousp, don't they?

B = A^2 seem to have a correspondence with $\Bbb Z_2$

let's look at the picture :

Alizter

How can I create a circle for it to slide on?

Balarka Sen

@Alizter bah it's right there

above. there is a picture of a circle.

then right click and vanish the points you don't want to appear in your work

uncheck the show object box, i.e.

in any case, the picture above should be given some importance. there are two sheets, as you see and cut and pasted togather through the imaginary real axis

@Alizter here's how it's relevant : take two sheets, corresponding to two copies of complex plane

Alizter

ok

Balarka Sen

now let $\sqrt{z}$ and $-\sqrt{z}$ act on the two sheets

the first on the bottom, the next on the upper

Alizter

15:41

yup

Balarka Sen

we branch the two sheets at $0$, as $\sqrt{0} = -\sqrt{0}$

i.e., paste the two sheets at the point $s = 0$.

Alizter

Yup

Balarka Sen

@Alizter now the two functions are both discontinuous in their sheets, when $z$ moves from the upper left sheet to the lower left sheet.

^ the pic on the right

Alizter

alright

Balarka Sen

to preserve continuity, one possible solution would be to cut both sheets through the negative real axis, and paste the A part with A' part, and the B part with B' part.

Alizter

15:45

yep

Balarka Sen

this gives you the surface at the left

of course, intersection happens when this is done in R^3 but it can be avoided in R^4

@Alizter now, let $z$ be a point on the bottom sheet on the surface on the left

Alizter

ok

Balarka Sen

for $z$ to loop once around the origin, it needs to go above to the upper sheet and then come back, right?

as there are pastings through the negative real axis/

Alizter

ok

Balarka Sen

but that, when seen from above by a flying kite, say, it'd look as if you are revolving twice around 0!

that is precisely what is happening in geogebra with A and B = A^2!!!

Alizter

15:49

ohh

interesting

so now Galois theory what does this mean?

Balarka Sen

so the surface on the right is the surface on which $\sqrt{z}$ is continuous.

it is not C, but a subset of C^2!

@Alizter coming to the point. first, rigorously the surface is $\{(a, b, c, d) \in \Bbb R^4 : (a + ib)^2 = c + id\}$.

to verify this, cross-section the surface.

Alizter

ok

Balarka Sen

you'll get $y = x^2$, the standard graph

^

Alizter

you like that picture :P

Balarka Sen

well it's intuitive :P

and i ain't have anymore picture of riemann surface of that!

Alizter

15:51

@BalarkaSen I cut one out of paper infront of me

Balarka Sen

LOL

Alizter

proceed

Balarka Sen

@Alizter so you wanted connection with galois theory, right?

Alizter

yup

Balarka Sen

in the graph A and B = A^2 in geogebra, B must loop twice for A to come back, right?

Alizter

15:54

ye

Balarka Sen

that relates to the cyclic group of order 2

Alizter

Was about to type that

now what about 4?

Balarka Sen

in other words, $w^2 - z$ has corresponding group $\Bbb Z_2$

@Alizter B = A^4?

Alizter

Yeah is it Z/4Z or Z/2Z^2?

Balarka Sen

Z/4Z

Alizter

15:55

how can we tell?

or rather construct klein 4?

Balarka Sen

just draw it out in geogebra to verify

Alizter

@BalarkaSen What polynomial has klein 4?

Balarka Sen

that i'd leave you to figure out.

just to add, these groups are called monodromy groups

and in fact it can be proved that monodromy groups = galois groups

'=' is here the isomorphism

@Alizter now recall A and B = log(A)

Alizter

@BalarkaSen Hmm, how would geoegebra?

I think klein 4 would be $x^4+ax^2+c$ or something

Balarka Sen

more or less. just make a good guess from galois theory

what is an extension of Q which has galois group Z_2^2?

Alizter

16:01

Q(i)?

wait

Balarka Sen

that'd have gal group Z_4

so no

Alizter

ahh I have not done this in a while

Balarka Sen

how about Q(2^1/2, 3^1/2)? does that work?

Alizter

yeah

Balarka Sen

then take, say, B = (A^2 - 2)*(A^2 - 3)

the two branch points are 2 and 3 on the complex plane

it'd be interesting to play with it

fiddle on! =)

Alizter

16:06

It cycles every two

Balarka Sen

oops i mean branch points are sqrt(2) and sqrt(3)

Alizter

that is weird and cool

What do symmetric groups look like

Balarka Sen

haha i see you've got a hang on it

Alizter

S_5

Balarka Sen

@Alizter ah, that

Alizter

16:09

x^5-x-1 i think has that

Balarka Sen

well B = A^5 - A - 1

Alizter

unless i cant remember

oh

Balarka Sen

yes, indeed

Alizter

yes

Balarka Sen

@Alizter ok i have to go. fiddle with it

just before i leave,

what i explained to you is called topological galois theory, in it's full rigorous form

Alizter

16:10

It makes a flower :D

Balarka Sen

it can actually be used to prove abel-ruffini theorem in a totally different way. using topology and geometry.

Alizter

If you look at $x^5-x-1$ on the unit circle

Balarka Sen

Indeed

Alizter

it plots a flower with 4 petals

Balarka Sen

The riemann surface is complicated

Alizter

16:11

thats valentines day sorted. Here you go S_5

Balarka Sen

hahaha

ok i am out

Alizter

sadly only 0.1% of the female population would get it

Balarka Sen

have fun with it

Alizter

@BalarkaSen Thank you.

Transcript for

$Mathematics$ Number theory