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

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A Gaussian unit is a Gaussian integer \(z\) such that \(\dfrac{1}{z}\) is also a Gaussian integer.
Well, we essentially want to show that \(\pm i,\pm 1\; |\; 1\) and they are the only ones.
But how will I show that they are the only ones?

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Other answers:

@Opcode and @experimentX any ideas?
No .. have to see.
Well... \(a^2 + b^2 = 1\)
I think you don't mean you mean
I mean Gaussian units. See my first post
Sorry! Opcode is still learning trigonometry. No "Gaussian units" for now :( Mabye @Hero ?
looks like there is solution on M.SE
OK yeah, I mean that they are a unit.
units of Gaussian Integer ... looks like related to ring theory.
ah! nice
What math is this? I see on the MSE thing it says: abstract-algebra, ring-theory. So is it both?
Ring theory is a subset of abstract algebra. This is theoretical mathematics also known as pure mathematics.
What would you even need this for? Like job wise?
@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
@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.
@experimentX Can we call this complex analysis?
Ouch. So much things to learn -_- If I was to attempt to get a Ph. D I'd be at age 40 :(.
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.
I think abstract algebra is at more fundamental level than complex analysis.
How old are you experimentX? (If I may ask?)
celebrating 25th spring
Wow. You guys are so smart! I feel so dumb around you guy :(
Just below average old guy!! There is nothing wow about me ... Looks like parth will grow up to be smart guy.
Yes, very smart indeed. @ParthKohli you education is so beautiful :-)
Aw, no! I will grow up to be some old-street beggar. You, my friend, have a huge market value. YOU SMARTYPANTS PROGRAMMER!
That's a pretty old screenshot. :-)
Lol yeah i found it on google images
yeah, they had written a blog post on me. -_-
Yeah I see its about how young you are haha
@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 :-)
@SWAG It's about how young and stupid I am. :-)
@Opcode No! There are people much better than me: @DLS and @shubhamsrg
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.
Wow, you could do trig?! WOW!
simple trigonometry ... that's the best i could do.
Pssh, tell the truth: Look at Parth: I can do simple trigonometry... :3 Parth taught me the basics :P I found the rest by internet :D
Juzt sho berry Amaysing! :O I am going to create a parth kohli fan club on facebook now.
Yush ^_^. I need to start using Facebook more anyway :-) Onward!
Just a joke Parth :-) I'm wasting away in class :| I really should be listening >.>
Not joining that group... When Parth says "..." It means --> No. Just No. *Any then the Parth meme was born :O*
Proceed, my child.
Boop ^.^
Don't. Start. Parth. Kohli. Memes. I'm. Not. Chuck. Norris.
Well I don;t dare to offend The Parth Kohli.
My apologies.
Opcode promises this is the last one: Now back to studying -_-
Would you like a shubhamsrg meme :3
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.
Next time if anyone calls you dumb use the ParthKohli defense!
ofcorse , as I said, I am gonna show him this screenshot! :)
Because when he says it, it can't be wrong!
Not that I am agreeing what he is saying, but he has said it, so no one can argue.
Hmm. Can I possible get a Ph. D in history? Or is that a no?
Only one person can answer all the questions in the world! Yes thats right,! We all know who that is. ^_^

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