Question

Show that \(1\), \(-1\), \(i\) and \(-i\) are the only Gaussian units and that there are no other Gaussian units.

Answer 1

A Gaussian unit is a Gaussian integer \(z\) such that \(\dfrac{1}{z}\) is also a Gaussian integer.

Answer 2

Well, we essentially want to show that \(\pm i,\pm 1\; |\; 1\) and they are the only ones.

Answer 3

But how will I show that they are the only ones?

Answer 4

@Opcode and @experimentX any ideas?

Answer 5

No .. have to see.

Answer 6

Well... \(a^2 + b^2 = 1\)

Answer 7

I think you don't mean http://en.wikipedia.org/wiki/Gaussian_units you mean http://en.wikipedia.org/wiki/Gaussian_integer

Answer 8

I mean Gaussian units. See my first post

Answer 9

Sorry! Opcode is still learning trigonometry.
No "Gaussian units" for now :(
Mabye @Hero ?

Answer 10

looks like there is solution on M.SE
http://math.stackexchange.com/questions/108071/unique-units-of-gaussian-integers

Answer 11

OK yeah, I mean that they are a unit.

Answer 12

units of Gaussian Integer ... looks like related to ring theory.

Answer 13

ah! nice

Answer 14

What math is this?
I see on the MSE thing it says:
abstract-algebra, ring-theory.
So is it both?

Answer 15

Ring theory is a subset of abstract algebra.

This is theoretical mathematics also known as pure mathematics.

Answer 16

What would you even need this for?
Like job wise?

Answer 17

@Opcode that is why they say that among a Ph.D in physics, a Ph.D in CS, a Ph.D in theoretical math and a large pepperoni pizza, the most different thing is a Ph.D in theoretical math.

The other three can feed a family of four X-D

Answer 18

@Opcode the abstract algebra is the generalization of the algebra we do. They make use of properties of these algebraic structures to solve the problem than we solve things directly.

Answer 19

@experimentX Can we call this complex analysis?

Answer 20

Ouch. So much things to learn -_-
If I was to attempt to get a Ph. D I'd be at age 40 :(.

Answer 21

I think you can't .... I haven't so far explored the depths of complex analysis. But I know these types of things get's mumbled-jumbled up.

Answer 22

I think abstract algebra is at more fundamental level than complex analysis.

Answer 23

How old are you experimentX? (If I may ask?)

Answer 24

celebrating 25th spring

Answer 25

Wow. You guys are so smart!
I feel so dumb around you guy :(

Answer 26

Just below average old guy!! There is nothing wow about me ... Looks like parth will grow up to be smart guy.

Answer 27

Yes, very smart indeed. @ParthKohli you education is so beautiful :-)

Answer 28

Aw, no!

I will grow up to be some old-street beggar. You, my friend, have a huge market value. YOU SMARTYPANTS PROGRAMMER!

Answer 29

http://blog.openstudy.com/wp-content/uploads/2012/06/Parth-SmartScore-Profile-copy.png

Answer 30

...?

Answer 31

That's a pretty old screenshot. :-)

Answer 32

Lol yeah i found it on google images

Answer 33

yeah, they had written a blog post on me. -_-

Answer 34

Yeah I see its about how young you are haha

Answer 35

@ParthKohli I'd rather be dumb in programming and have your math skills.
Heck I'd even drop studying history.
@ParthKohli is my role model :-)

Answer 36

@SWAG It's about how young and stupid I am. :-)

Answer 37

@Opcode No! There are people much better than me:

@DLS and @shubhamsrg

Answer 38

looks like @ParthKohli is being modest ,,, don't you go to school? When i was thirteen i was in eighth grade. I could do simple trigonometry.

Answer 39

Wow, you could do trig?! WOW!

Answer 40

simple trigonometry ... that's the best i could do.

Answer 41

Pssh, tell the truth:
Look at Parth:
http://blog.openstudy.com/2012/06/27/our-youngest-user-and-quickest-to-99/
I can do simple trigonometry... :3
Parth taught me the basics :P
I found the rest by internet :D

Answer 42

Juzt sho berry Amaysing! :O

I am going to create a parth kohli fan club on facebook now.

Answer 43

Yush ^_^.
I need to start using Facebook more anyway :-)
Onward!

Answer 44

...

Answer 45

(Y)

Answer 46

Just a joke Parth :-)
I'm wasting away in class :|
I really should be listening >.>

Answer 47

http://www.facebook.com/groups/499948566734706/

Answer 48

Not joining that group...
When Parth says "..."
It means --> No. Just No.
*Any then the Parth meme was born :O*

Answer 49

Proceed, my child.

Answer 50

Boop ^.^
http://cdn.memegenerator.net/instances/400x/35373413.jpg

Answer 51

Don't.
Start.
Parth.
Kohli.
Memes.

I'm.
Not.
Chuck.
Norris.

Answer 52

Well I don;t dare to offend The Parth Kohli.

Answer 53

My apologies.

Answer 54

Opcode promises this is the last one:
http://cdn.memegenerator.net/instances/400x/35373583.jpg

Now back to studying -_-

Answer 55

http://gyazo.com/73d47564ed85f1b06dcf4d05f3a8eae6

Answer 56

Would you like a shubhamsrg meme :3

Answer 57

I am gonna set that image as my desktop wallpaper, my room wallpaper, even gonna get it tatooed . Gonna preserve its print out like a certificate.. The Parth Kohli said that! wow! that is gonna help me in my future.

Answer 58

Next time if anyone calls you dumb use the ParthKohli defense!

Answer 59

ofcorse , as I said, I am gonna show him this screenshot! :)

Answer 60

Because when he says it, it can't be wrong!

Answer 61

Not that I am agreeing what he is saying, but he has said it, so no one can argue.

Answer 62

Hmm. Can I possible get a Ph. D in history?
Or is that a no?

Answer 63

Only one person can answer all the questions in the world!
Yes thats right,! We all know who that is. ^_^