Mathematics - 2012-03-12 (page 1 of 3)

t.b.

00:00

(so no need to resort to pastebins)

I don't think there's a fail-safe general recipe for determining limit points. You need to play around with the set at hand and see if you can say something sensible. In the case at hand, you should be able to come up with a conjecture. To prove it, you need to estimate a little.

arete

I apologize but I do not know latex. I mean it is evident that that the set goes to zero pretty quickly. I do not know how to go about showing that this is the only limit point though or determining if it is.

t.b.

So, the numbers you have are all positive, right? And when you say "it goes to zero pretty quickly" the intuition is: no matter what m is the quotient F_m / F_n is very small since m < n. Can you quantify that?

In other words: can you give an upper bound of F_m / F_n in terms of n and m?

arete

The first term (m=1 and n=2) would be our upper bound because 1 is the smallest difference we can have between n and m and therefore n=2 is the smallest denominator we can have.

t.b.

00:16

Yes, that's right. Now try to adapt this idea to show that there are only finitely many quotients that are larger than any given positive epsilon. Do you see what this would give?

arete

This would give that zero is an accumulation point. Would that simultaneously show that zero is the only accumulation point?

t.b.

Yes, because there's nothing a finite set can accumulate to!

arete

Ohhh!

If there was any doubt I could assume that there were two accumulation points and this would lead to a contradiction.

t.b.

So, you've essentially convinced yourself that 0 is an accumulation point and now you should make the idea that no other one can be an accumulation point precise. To achieve this, look at the fractions and estimate them from above (by making the numerator larger and the denominator slightly smaller). This should give you a good enough bound to show what I asserted earlier on.

It's getting late here, so I should go to bed now. I'm pretty confident that you'll tackle that problem. Good luck and see you around!

Good night y'all

arete

Thank you tb! I really appreciate your help.

t.b.

00:42

Gee, Kannappan! Go to bed!

ymar

01:15

Is anybody here?

Kannappan Sampath

Yes, me.

ymar

Hi! I've been wanting to ask you something.

Is your avatar random?

Dennis Hayden

scan it

then u will know :D

ymar

What does it mean?

Dennis Hayden

it's a qr code or it looks like one.

ymar

01:18

I have no idea what a qr code is...

Kannappan Sampath

@DennisHayden Yes, it is a QR code.

@ymar Wikipedia. : )

ymar

I just found the same picture on the internet

Dennis Hayden

I f i had to guess it is realy and it links to a site that @KannappanSampath like/wants people to see.

ymar

Oh, I see.

It's on that wikipedia page too.

Kannappan Sampath

@ymar Not the same, but they look similar. You cannot even guarantee a non-trivial homomorphism I say.

Hi @Sivaram

Sivaram Ambikasaran

01:19

Hi @KannappanSampath

did u see the chat I sent on skype

I started reading Halmos

ymar

Anyway, does anybody know how I can access an obscure doctoral dissertation from 1969?

Sivaram Ambikasaran

it is very good to read and understand the underlying concepts. I read the first two sections and have hopefully clearly inderstood the motivation for Axiom of extension and axiom of specification

@ymar Can you give us the title of the doctoral dissertation?

ymar

It's here. The Multiplicative Semigroup of a Ring

The best mathematical library in my city doesn't have it.

Kannappan Sampath

@ymar Is this link useful.

lib.utk.edu/refs/dissertations.html

Dennis Hayden

01:37

I have a question for the function Q($x$)= $x^2+c$ . How do I find the value of c that have a super-attracting fixed point?

ymar

@KannappanSampath Yes! Apparently, they sell copies. It would be a drastic measure, but I may have to do it.

Kannappan Sampath

Well, think about asking your theses advisor to write to someone there in UTK and ask for a copy of the theses. It should be possible.

ymar

For free? That would be good. I've just found out that it costs $49. I've never bought a book for this kind of money! :)

Kannappan Sampath

49$ stabs my gut already! : (

Dennis Hayden

So anyone have an idea to find valuse of c with a super attracting fixed points? :D

ymar

01:48

What does super-attracting mean?

Gee. There's too much stuff on the internet.

anon

"too much stuff on the internet" - lemme guess, you googled "super-attracting" :)

ymar

That too. I google too much.

Dennis Hayden

Fixed point (mathematics)

:Not to be confused with a stationary point where f(x) = 0. In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set. That is to say, c is a fixed point of the function f(x) if and only if f(c) = c. For example, if f is defined on the real numbers by :\ f(x) = x^2 - 3 x + 4, then 2 is a fixed point of f, because f(2) = 2. Not all functions have fixed points: for example, if f is a function defined on the real numbers as f(x) =...

ymar

@DennisHayden There's no definition of super-attracting there...

Dennis Hayden

yea but it explains what a fixed point is

Kannappan Sampath

01:53

@anon Do you use MikTeX, please say yes. : )

ymar

:-)

Dennis Hayden

or what an attracting fixed point it.

anon

yes.

Kannappan Sampath

Now, shall I ask for help in using some new packages I installed, please? @anon

ymar

Alright, I need to go to bed. People should stop writing all those papers and get a life.

Goodnight.

Dennis Hayden

01:56

good night

anon

The only package I installed, I put the file somewhere in a MikTeX folder and then just put \usepackage in my header. And that is the extent of my knowledge of installing packages :D

Kannappan Sampath

:/ I need help badly, very badly.

Dennis Hayden

I can try to help I am a computer ninja.

Kannappan Sampath

@DennisHayden Well, I installed a few packages and when I say use package, it shows some error saying .sty file not found. What do I do. then?

Dennis Hayden

do you know where you installed the packages and where the program read/looks for the files.

Sivaram Ambikasaran

02:02

googel for the .sty file

and copy it in the same folder

as your tex file

it should do the job

Kannappan Sampath

Oh, I see. Let me try now.

leo

@KannappanSampath do you use MikTeX or TeXLive?

Kannappan Sampath

@leo MikTeX

leo

@KannappanSampath 1.Go to the MikTex Settings(Admin)

Kannappan Sampath

Ok.

leo

02:17

@KannappanSampath 2. In the tab "General", in the option "Instal missing packages on-the-fly" select "Ask me first"

3. Then go to the tab "Packages" (that can take few seconds)

Mariano Suárez-Alvarez

Am I special or a few things got broken on the main site?

Nothing happens if I click in the inboxy thing on the top left corner, for example

anon

works fine for me

Mariano Suárez-Alvarez

I am special!

leo

@KannappanSampath 3. Click "Change..." and select "Packages shall be instaled from the internet" and click "Next"

4. Select your preferred repository

5.Then OK and OK

Kannappan Sampath

@leo FYI, I already installed the package. Do I still have to go through step 3?

leo

02:23

@KannappanSampath wait a second

when you do all this

you compile your document and, if everything is fine

pops up a window that ask you if you want to compile your document

then you accept

and it works

If you have already installed the package then perhaps your IDE don't know where are the directories of LaTeX

Kannappan Sampath

Going through step 5. now.

leo

what do you use? Texmaker?

Kannappan Sampath

Yes. Texmaker. Is that a wrong thing to use?

(Seriously, I am facing hiccups with LaTeX like hell.)

leo

@KannappanSampath no. Is the better. I think :-)

what package is? by the way

Kannappan Sampath

exercise package.

leo

02:29

@KannappanSampath do you installed it with the wizard provided by MiKTeX?

Kannappan Sampath

@leo Yes. I'll plug in LaTeX error here.

! LaTeX Error: File `exercises.sty' not found.Type X to quit or <RETURN> to proceed,or enter new name. (Default extension: sty)Enter file name:! Emergency stop.<read > \title

Rajesh D

even I use Texmaker

leo

@KannappanSampath do you configured MiKTeX as I have recommend?

Kannappan Sampath

@leo Yes. :-)

leo

ok

fabianhjr

02:32

I was given a weird problem that I don't understand. It states something along the lines of I have a pet with a head of 16 cm, a tail of 16 cm + 1/4 the size of the body(trunk) and the trunk is equal to the sum of the head and the tail. Isn't this recursive?

leo

@KannappanSampath Go to the Texmaker/Options/Commands

and check all the paths

Rajesh D

@Kannappan : Placing the file exercise.sty (style file) manually in the working directory would solve the problem

Kannappan Sampath

You mean, configure TeXmaker...?

leo

@KannappanSampath yes

Kannappan Sampath

@RajeshD How would I do that? This seems to be precisely what I want to know. : )

leo

02:36

@RajeshD not if the package needs other files than the .sty, as usual

Rajesh D

Where is your current latex file which you are making is the working directory....now place the exercise.sty in tis folder.....not only this all the other files which came with your excersise should be placed here

leo

for example in front of "LaTeX" must be something like
`"C:/texlive/2011/bin/win32/latex.exe" -interaction=nonstopmode %.tex`

without the ` `

Kannappan Sampath

@leo Those are all fine. Because others compile. Just those docs that use new packages don't.

Rajesh D

Do you have the file exercise.sty with you @Kannappan

?

Kannappan Sampath

@RajeshD Yes. Thanks to @Sivaram, he sent me the file. :-)

leo

02:40

@KannappanSampath ok. I see the problem. wait a second

Rajesh D

now What ever Sivaram has sent you...put all those files in one directory.....now work in that directory..ie place the current latex file you are making in that directory

fabianhjr

Ok, wolfram solved my issue. wolframalpha.com/input/… I still do not understand it.

Rajesh D

make sure your file name is same as that of the file name given by Sive...take a back up of it...then open it and remove the matter in it and start typing your matter

Kannappan Sampath

@RajeshD He sent me only one file. (the .sty file) and I have a .tex file I put all of them in a folder in desktop. Is this right?

leo

@KannappanSampath try to compile a document with the command \usepackage{subfig}

Rajesh D

02:42

yes

leo

@KannappanSampath tell me if pop up a window that ask you if you want to install that package

Rajesh D

you need to mention in our .tex file directing to use the specified .sty file

Kannappan Sampath

@All Miraculously, TeXmaker behaves sensibly. I am through. Done!

It works.

leo

what do you do

Kannappan Sampath

I should thank all of you for all the help.

@Sivaram Thank you for .sty file.

leo

02:46

you move the file to you work directory

Kannappan Sampath

@Rajesh and @leo For the instructions.

leo

@KannappanSampath try with \usepackage{thmtools}

Kannappan Sampath

@leo Should I install that package.

leo

@KannappanSampath no. Is Just to prove if he ask you, if you want to install it

t.b.

sheesh, still not sleeping, Kannappan?

Kannappan Sampath

02:52

Well I am off to sleep people.

@tb I am going to sleep right away. :-)

anon

Isn''t it like 10am where kanna's at?

Jeff

hi folks. i guess i left this browser window open when i shut down :D

...and I see many of the same people are still here! :D

Rajesh D

04:14

yes @anon now it is

Manishearth

Hello everyone! Does anyone have a link to your discussion to enable mathjax on meta? I can find the mathjax-on-main discussion. Or did you get mathjax support simultaneously for both sites? On physics.SE we have mathjax on the main site but not on meta, and we want to see if its easy to acquire...

leo

you want to the meta or the chat

Manishearth

Meta

leo

i think you should request that to the moderators

Manishearth

Oh, I was discussing it with a mod

We wanted to see how hard it was to get mathjax

Manishearth

04:53

Basically we wanted to see if it was worth the effort..

5T0M

08:34

how do we type mathematical symbols here

do we need to install some special software

special symbols like integreal signs

integral

anon

@5T0M: Have you heard of LaTeX?

5T0M

Yes little bit

anon

We use it in chat here. See math.ucla.edu/~robjohn/math/mathjax.html

5T0M

so we have to learn this language to be able to write symbols?

anon

That or you could just try to use plain English or ascii text. If you want help with latex there's this and this

Daniil

09:14

Hello

Ilya

@Dan: hi

5T0M

THANKS for the help @anon

Jonas Teuwen

09:35

@Ilya A colleague of mine has her PhD defense at 12h :-).

anon

10:04

Jonas Teuwen

@anon New Age LaTeX usage.

5T0M

i.stack.imgur.com what type of site is this?

is it owned by stack exchange?

anon

In the sense that SE bought its usage (which I assume is the case by "commercial"), yes.

5T0M

oh nice

are you from ucla @anon

anon

10:21

no

5T0M

what was this link about? math.ucla.edu/~robjohn/math/…

i couldnt undestand what is to be done once the page is open

math.ucla.edu/~robjohn/math/mathjax.html

anon

do you know about your browser's bookmark tab?

5T0M

iya but apart from that

line number 4 and 5 are latex , i can make that out,

anon

What are you talking about? You click and hold on the blue "render mathjax" link, drag it to your bookmarks bar, and then let go. A bookmark for "render mathjax" will be created, and every time you click it your browser will turn any LaTeX inside dollar signs into the actual symbols. What part of this are you having problems with?

Also, just because someone links to an educational establishment doesn't mean they are affiliated with it. You'll notice the url is for someone named "robjohn," who happens to correspond to that orange mean-square avatar at the top right of the chatroom (currently idle).

5T0M

haha yeah

so it will only help me to read Latex, and not write it

oh its happening, the dollars signs are converting

thanks man

but no bookmark has been created in my bookmarks bar?

anon

10:39

where did you drag the "render mathjax" to in your browser? what browser do you use?

5T0M

i dragged it to the address bar, i use chrome

anon

the address bar =/= the bookmarks bar

(not the same thing)

5T0M

there is no bookmarks bar

or is there?

anon

click the wrench button in the top right, then hover over "Bookmarks >" and a menu will appear. Click "Show Bookmarks bar"

5T0M

got it

thansk

thanks

Ilya

12:06

@JonasTeuwen oh, I've missed it :(

@Matt: hello

Daniil

13:07

Sup @Ilya

Uhm, it's kinda quite in here today, isn't it?

anon

quiet, and there are always quiet periods in the day. in the last week I think I witnessed a 6 hour period of silence.

or maybe that was last week, or whatever. who knows.

N3buchadnezzar

13:38

Heya !

Anyone know how I can draw an area expressed by

$1>z>0 \,,\, 0<y<x \,,\, z>x>0$

? =)

anon

that would be a volume, not an area

Daniil

well, if you fix $z$, then you have an area under a triangle: $(0,0), (z, 0), (z,z)$

Jonas Teuwen

14:08

@Ilya Back! Did you ever see a Delft PhD defense?

Daniil

14:40

Is it the case that $L^{\omega} = L^* \cap \Sigma^{\omega}$?

(in the context of $\omega$-languages)

Srivatsan

16:35

hi tb

t.b.

Hey @Srivatsan looks like you showed up on one of the most inactive days in a while :)

How are you?

Kannappan Sampath

Hi @Srivatsan @tb

t.b.

Hi Kannappan, did you sleep at least a little?

Srivatsan

@tb That is not too bad. I have to leave in a few minutes anyway =)

hi Kannappan

Kannappan Sampath

A lot more. Actually, there were no classes today, from afternoon, till 6, I had slept enough. @tb

t.b.

16:37

@Srivatsan I see. Glad to meet you even though it's only gonna be a short meeting

@KannappanSampath great! I'm proud of you

Srivatsan

I will continue to drop by every once in a while

Kannappan Sampath

@tb Hey, joking, uh?

t.b.

@Srivatsan Good to hear. I'll have to cut down my presence here again :)

Kannappan Sampath

@tb Why, again? So, more people will be on leave from MSE. </3

Srivatsan

@tb Ah, frankly, that might not worry me too much :=) .

Ilya

16:42

@Srivatsan: hi, nice to meet you again

Srivatsan

Ilya, hey hi

Ilya

@Jonas: yes,I've already seen it, what about?

Srivatsan

Good to run into you

Ilya

:) grading exams one more day

@Kan hello

Jonas Teuwen

@Ilya Oh nothing :-).

Ilya

16:42

@tb: hi, nice example, thanks :)

Kannappan Sampath

@Ilya Hi!

@Jonas How are the integrals doing?

Srivatsan

Maybe it's a good time to mention:

Jonas Teuwen

@KannappanSampath I am doing them as we speak.

Computing them I mean.

Srivatsan

It's my birthday tomorrow.

Ilya

nice, we'll bake you a cake

Kannappan Sampath

16:44

@Srivatsan Many More Happy returns of the day.

Srivatsan

Thanks, Kannappan

Jonas Teuwen

So you're going to be old too? Like me.

Srivatsan

@Ilya Ah, cake. Super. :)

Ilya

@Srivatsan: oh, you'll be as old as me! :) Let me send you a list of pills we take in 24 ;)

@KannappanSampath tomorrow, not today. Hide it from him :)

Kannappan Sampath

(The cake from TeX.SE will appear tommorrow, here.)

It's not long. Like two hours from now in India. (May I ask you hang around here until then?)

Srivatsan

16:47

@Ilya Might be useful. Now I understand what they mean: probably my list of minor complaints with my body has become significantly longer in the last few months

:)

Ilya

@Srivatsan :)

Srivatsan

@KannappanSampath Would like to. But my brother will soon occupy the room, so I can't.

Kannappan Sampath

@Srivatsan </3

Rajesh D

Hi @Sri : How r u doin

Srivatsan

@KannappanSampath hmm, sometimes I do wish I were the elder one. :/

@RajeshD hi Rajesh, how are you?

I am good

Rajesh D

16:51

fine...good to see you...are you in the subcontinent or back to US

?

Kannappan Sampath

@Srivatsan Like I am in my family. :-)

Srivatsan

@KannappanSampath Oh, are you? Do you have any siblings?

Kannappan Sampath

@Srivatsan Yes, I have a brother, younger one. :-)

Srivatsan

@RajeshD I am in the subcontinent.

Rajesh D

Oh good......hot wheather in Chennai ?

Srivatsan

16:53

@RajeshD Not very hot yet, but yes.

Kannappan Sampath

Thinking of which, I will have to be in Chennai, this summer, in such a hot weather.

Srivatsan

How come?

Kannappan Sampath

For the SRFP thingie at IMSc. :-)

Srivatsan

SRFP: I don't know what this is.

Kannappan Sampath

Ah, never saw that. ^

SRFP- Summer Research Fellowship Programme by the Indian Academy of Sciences.

Srivatsan

16:56

Nothing. I asked if you are talking about spending your summer at your home in Chennai.

I wasn't sure where your parents stay.

Kannappan Sampath

@Srivatsan In Trichy.

Srivatsan

Oh cool

Alright, I have to go now. See you guys. Maybe I will be back tomorrow.

Jonas Teuwen

See you!

Kannappan Sampath

Bye Srivatsan. Have a wonderful day tommorrow. And, here is a small gift I had already given you.

Matt N.

Hi and bye, @Srivatsan!

Srivatsan

16:59

Thanks.

Kannappan Sampath

Rajesh D

bye

Kannappan Sampath

Hi @Matt

(The proof about equivalence relation looks good.)

Matt N.

Hi Kannappan!

Thanks : )

Oh dear! 6 stars. Do I have to read this as "Yes, you definitely have to get rid of this image" or do I read this as "Aww cute"?

Kannappan Sampath

@MattN Which one do you mean?

t.b.

17:02

@Srivatsan Bye Srivatsan! Have a nice birthday tomorrow!

@Ilya what do you mean? The Tietze extension?

Hi Matt

Matt N.

@KannappanSampath There's only one with 6 stars at the moment, no? : )

Hi teddy. tuts at teddy

t.b.

what's up?

Matt N.

You were. All night. </3

Kannappan Sampath

@MattN Yes. And, no. I thought you posted some image and it's already gone from the starboard. :-)

Matt N.

: )

t.b.

17:05

@MattN that's not true... I slept from about 4 to 10, so plenty enough :)

Jonas Teuwen

Hmm, by computing a lot of integrals you become better at computing them (no shit Sherlock).

Matt N.

@tb Oh, then I jumped to conclusions (again) : )

@JonasTeuwen Can you also compute infinite sums of sines and cosines? I'm asking because I can't really do that and I want to see how that works.

Jonas Teuwen

Compute as in give a formula in terms of known functions and no series?

You can of course define a function... that is your series.

Matt N.

@JonasTeuwen Probably...

Jonas Teuwen

So you should specify what you mean with "compute".

So you have stuff like $\sum a_n \cos(n x) + b_n \cos(n x)$?

Matt N.

17:08

I would assume it turns out to be e to the something or some other easy to write down function.

Jonas Teuwen

In general: No.

I can of course define the Matt-Jonas function $M(x)$ 8-).

t.b.

@MattN the problem is that this domain is infested by other sciences, so it's hard to find a reasonable source for it...

Matt N.

I have
$$ \sum_{n = - \infty}^\infty \frac{\sin (\frac{3}{4} \pi n )}{ \pi n} (\cos (- \omega n) + i \sin (- \omega n))$$ : )

Jonas Teuwen

@tb Sorry? I don't understand :-).

@MattN The horror!

Rajesh D

The Jonas brothers @jonas were you in it ?

Matt N.

17:11

@JonasTeuwen That's what I thought so I looked up the solution. (Rectangle function)

Jonas Teuwen

@RajeshD Not sure what they are but yes.

t.b.

@JonasTeuwen The chat kept on moving upwards. It was intended for Matt

Jonas Teuwen

@MattN So you can actually compute that sine right?

Matt N.

@tb Yes. Such as engineering (computer science) and signal processing.

Jonas Teuwen

Do you have something against signal processing?

Working on that.

Matt N.

17:12

@JonasTeuwen You should specify what you mean with "compute".

Jonas Teuwen

@MattN Give a numerical value.

Kannappan Sampath

Now, no fun. I should sleep today, or not attend classes tommorrow. Looking at the consequences of not attending classes, it occurs to me that I must sleep. Bye people.

Jonas Teuwen

Then mess a bit with the subseries...

Matt N.

@JonasTeuwen Right. Unfortunately $n$ isn't a constant.

Jonas Teuwen

@KannappanSampath Good night ding ding ding ding!

@MattN Sure, but it can only be: $\pm 1, 0, \pm 2^{-\frac12}$.

jay wendt

17:13

I'm new to Galois Theory, trying to pick it up. I'm trying to determine the Galois group of $f(X)=X^n-t$ over $\mathbb{C}(t)$, for $t$ transcendental over $\mathbb{C}$. I think it's obvious that the Galois group of $f$ over $\mathbb{C}$ is cyclic of order $n$, but I'm not sure how to think of the Galois group of $f$ over $\mathbb{C}(t)$.

Matt N.

@KannappanSampath Sleep well!

t.b.

@KannappanSampath good night!

5T0M

DOes anyone know of Riff and ROtes?

Jonas Teuwen

Hmm, I have a series of hypergeometric series... 8-).

Matt N.

@JonasTeuwen I see what you're saying. Although I get only 4 different values. Thank you!! : )

Jonas Teuwen

17:17

I should be able to compute that.

@MattN Which one did I give too much?

$-1$?

Matt N.

I haven't worked out which but I get 3/4 , 6/4 , 2 1/4, 3

Rajesh D

Jonas Brothers >>

Jonas Teuwen

Hmmm.

Rajesh D

►

Matt N.

Let me see if I can manage without a calculator.

Rajesh D

17:18

this is a dummy video though

sorry for the distraction....i wasn't aware of the discussion goin on here

Matt N.

6/4 pi maps to -1

Then 1 must be wrong.

5T0M

Riff and Rotes , anyone, knows or is enthusiastic about, just let me know

jay wendt

I want to say that the Galois group of $f=X^n-t$ over $\mathbb{C}(t)$ is trivial since $1/t^n$ lies in $\mathbb{C}(t)$, as do all of the $n$-th roots of unity. Can someone tell me if I am right?

Matt N.

OTOH I get 0.25 pi $\simeq $ 0.75 pi both map to $\sqrt{2} / 2$.

jay wendt

i feel like i must be making some mistake here, it seems too trivial

Matt N.

17:25

@JonasTeuwen What do you do with $e^{- i \omega n}$?

Rajesh D

@STOM : What is riff and rotes ?

5T0M

@RajeshD its a consice way to write numbers

concise

wait I will just get the link

t.b.

@RajeshD check this

5T0M

here it is oeis.org/wiki/Riffs_and_Rotes

oh both are same links

@t

Rajesh D

@STOM : thanks for the link...i'll go through them....for the moment i need to attend something bbl

5T0M

17:35

@tb how did you make link out of the word this,

ok @RajeshD no worries

t.b.

@5T0M use [link text](http://...). See here for more.

By the way: you can edit your recent posts during two minutes by hitting the up arrow or by clicking at the leftmost end and selecting "edit"

5T0M

very thanks for that @tb

jay wendt

18:07

can anyone give me just a little direction?

t.b.

I have to go for a while. bbl.

Matt N.

See you later!

5T0M

18:34

bye everyone,,,,,,,,, have a good night

jay wendt

Ok, I was being very foolish. Obviously C(t) doesn't include any roots of t, so the galois group will be cyclic of order n

Transcript for

$Mathematics$ Mathematics