## ParthKohli 2 years ago Show that $$1$$, $$-1$$, $$i$$ and $$-i$$ are the only Gaussian units and that there are no other Gaussian units.

1. ParthKohli

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

2. ParthKohli

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

3. ParthKohli

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

4. ParthKohli

@Opcode and @experimentX any ideas?

5. experimentX

No .. have to see.

6. ParthKohli

Well... $$a^2 + b^2 = 1$$

7. experimentX

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

8. ParthKohli

I mean Gaussian units. See my first post

9. Opcode

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

10. experimentX

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

11. ParthKohli

OK yeah, I mean that they are a unit.

12. experimentX

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

13. ParthKohli

ah! nice

14. Opcode

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

15. ParthKohli

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

16. Opcode

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

17. 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

18. 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.

19. ParthKohli

@experimentX Can we call this complex analysis?

20. Opcode

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

21. 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.

22. experimentX

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

23. Opcode

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

24. experimentX

celebrating 25th spring

25. Opcode

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

26. experimentX

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

27. Opcode

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

28. ParthKohli

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

29. SWAG
30. ParthKohli

...?

31. ParthKohli

That's a pretty old screenshot. :-)

32. SWAG

Lol yeah i found it on google images

33. ParthKohli

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

34. SWAG

Yeah I see its about how young you are haha

35. 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 :-)

36. ParthKohli

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

37. ParthKohli

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

38. 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.

39. ParthKohli

Wow, you could do trig?! WOW!

40. experimentX

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

41. 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

42. shubhamsrg

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

43. Opcode

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

44. ParthKohli

...

45. shubhamsrg

(Y)

46. Opcode

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

47. shubhamsrg
48. Opcode

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

49. ParthKohli

Proceed, my child.

50. Opcode
51. ParthKohli

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

52. shubhamsrg

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

53. shubhamsrg

My apologies.

54. Opcode

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

55. shubhamsrg
56. Opcode

Would you like a shubhamsrg meme :3

57. 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.

58. Opcode

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

59. shubhamsrg

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

60. shubhamsrg

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

61. shubhamsrg

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

62. Opcode

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

63. shubhamsrg

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