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

- ParthKohli

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- ParthKohli

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

- ParthKohli

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

- ParthKohli

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

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## More answers

- ParthKohli

@Opcode and @experimentX any ideas?

- experimentX

No .. have to see.

- ParthKohli

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

- experimentX

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

- ParthKohli

I mean Gaussian units. See my first post

- Opcode

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

- experimentX

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

- ParthKohli

OK yeah, I mean that they are a unit.

- experimentX

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

- ParthKohli

ah! nice

- Opcode

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

- ParthKohli

Ring theory is a subset of abstract algebra.
This is theoretical mathematics also known as pure mathematics.

- Opcode

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

- ParthKohli

@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

- experimentX

@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.

- ParthKohli

@experimentX Can we call this complex analysis?

- Opcode

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

- experimentX

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.

- experimentX

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

- Opcode

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

- experimentX

celebrating 25th spring

- Opcode

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

- experimentX

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

- Opcode

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

- ParthKohli

Aw, no!
I will grow up to be some old-street beggar. You, my friend, have a huge market value. YOU SMARTYPANTS PROGRAMMER!

- SWAG

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

- ParthKohli

...?

- ParthKohli

That's a pretty old screenshot. :-)

- SWAG

Lol yeah i found it on google images

- ParthKohli

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

- SWAG

Yeah I see its about how young you are haha

- Opcode

@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 :-)

- ParthKohli

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

- ParthKohli

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

- experimentX

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.

- ParthKohli

Wow, you could do trig?! WOW!

- experimentX

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

- Opcode

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

- shubhamsrg

Juzt sho berry Amaysing! :O
I am going to create a parth kohli fan club on facebook now.

- Opcode

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

- ParthKohli

...

- shubhamsrg

(Y)

- Opcode

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

- shubhamsrg

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

- Opcode

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

- ParthKohli

Proceed, my child.

- Opcode

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

- ParthKohli

Don't.
Start.
Parth.
Kohli.
Memes.
I'm.
Not.
Chuck.
Norris.

- shubhamsrg

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

- shubhamsrg

My apologies.

- Opcode

Opcode promises this is the last one:
http://cdn.memegenerator.net/instances/400x/35373583.jpg
Now back to studying -_-

- shubhamsrg

http://gyazo.com/73d47564ed85f1b06dcf4d05f3a8eae6

- Opcode

Would you like a shubhamsrg meme :3

- shubhamsrg

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.

- Opcode

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

- shubhamsrg

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

- shubhamsrg

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

- shubhamsrg

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

- Opcode

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

- shubhamsrg

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

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