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ParthKohli

  • 2 years ago

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

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  1. ParthKohli
    • 2 years ago
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    A Gaussian unit is a Gaussian integer \(z\) such that \(\dfrac{1}{z}\) is also a Gaussian integer.

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

  3. ParthKohli
    • 2 years ago
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    But how will I show that they are the only ones?

  4. ParthKohli
    • 2 years ago
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    @Opcode and @experimentX any ideas?

  5. experimentX
    • 2 years ago
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    No .. have to see.

  6. ParthKohli
    • 2 years ago
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    Well... \(a^2 + b^2 = 1\)

  7. experimentX
    • 2 years ago
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    I think you don't mean http://en.wikipedia.org/wiki/Gaussian_units you mean http://en.wikipedia.org/wiki/Gaussian_integer

  8. ParthKohli
    • 2 years ago
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    I mean Gaussian units. See my first post

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

  10. experimentX
    • 2 years ago
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    looks like there is solution on M.SE http://math.stackexchange.com/questions/108071/unique-units-of-gaussian-integers

  11. ParthKohli
    • 2 years ago
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    OK yeah, I mean that they are a unit.

  12. experimentX
    • 2 years ago
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    units of Gaussian Integer ... looks like related to ring theory.

  13. ParthKohli
    • 2 years ago
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    ah! nice

  14. Opcode
    • 2 years ago
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    What math is this? I see on the MSE thing it says: abstract-algebra, ring-theory. So is it both?

  15. ParthKohli
    • 2 years ago
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    Ring theory is a subset of abstract algebra. This is theoretical mathematics also known as pure mathematics.

  16. Opcode
    • 2 years ago
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    What would you even need this for? Like job wise?

  17. ParthKohli
    • 2 years ago
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    @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
    • 2 years ago
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    @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
    • 2 years ago
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    @experimentX Can we call this complex analysis?

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

  21. experimentX
    • 2 years ago
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    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
    • 2 years ago
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    I think abstract algebra is at more fundamental level than complex analysis.

  23. Opcode
    • 2 years ago
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    How old are you experimentX? (If I may ask?)

  24. experimentX
    • 2 years ago
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    celebrating 25th spring

  25. Opcode
    • 2 years ago
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    Wow. You guys are so smart! I feel so dumb around you guy :(

  26. experimentX
    • 2 years ago
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    Just below average old guy!! There is nothing wow about me ... Looks like parth will grow up to be smart guy.

  27. Opcode
    • 2 years ago
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    Yes, very smart indeed. @ParthKohli you education is so beautiful :-)

  28. ParthKohli
    • 2 years ago
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    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
    • 2 years ago
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    http://blog.openstudy.com/wp-content/uploads/2012/06/Parth-SmartScore-Profile-copy.png

  30. ParthKohli
    • 2 years ago
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    ...?

  31. ParthKohli
    • 2 years ago
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    That's a pretty old screenshot. :-)

  32. SWAG
    • 2 years ago
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    Lol yeah i found it on google images

  33. ParthKohli
    • 2 years ago
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    yeah, they had written a blog post on me. -_-

  34. SWAG
    • 2 years ago
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    Yeah I see its about how young you are haha

  35. Opcode
    • 2 years ago
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    @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
    • 2 years ago
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    @SWAG It's about how young and stupid I am. :-)

  37. ParthKohli
    • 2 years ago
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    @Opcode No! There are people much better than me: @DLS and @shubhamsrg

  38. experimentX
    • 2 years ago
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    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
    • 2 years ago
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    Wow, you could do trig?! WOW!

  40. experimentX
    • 2 years ago
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    simple trigonometry ... that's the best i could do.

  41. Opcode
    • 2 years ago
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    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
    • 2 years ago
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    Juzt sho berry Amaysing! :O I am going to create a parth kohli fan club on facebook now.

  43. Opcode
    • 2 years ago
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    Yush ^_^. I need to start using Facebook more anyway :-) Onward!

  44. ParthKohli
    • 2 years ago
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    ...

  45. shubhamsrg
    • 2 years ago
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    (Y)

  46. Opcode
    • 2 years ago
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    Just a joke Parth :-) I'm wasting away in class :| I really should be listening >.>

  47. shubhamsrg
    • 2 years ago
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    http://www.facebook.com/groups/499948566734706/

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

  49. ParthKohli
    • 2 years ago
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    Proceed, my child.

  50. Opcode
    • 2 years ago
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    Boop ^.^ http://cdn.memegenerator.net/instances/400x/35373413.jpg

  51. ParthKohli
    • 2 years ago
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    Don't. Start. Parth. Kohli. Memes. I'm. Not. Chuck. Norris.

  52. shubhamsrg
    • 2 years ago
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    Well I don;t dare to offend The Parth Kohli.

  53. shubhamsrg
    • 2 years ago
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    My apologies.

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

  55. shubhamsrg
    • 2 years ago
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    http://gyazo.com/73d47564ed85f1b06dcf4d05f3a8eae6

  56. Opcode
    • 2 years ago
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    Would you like a shubhamsrg meme :3

  57. shubhamsrg
    • 2 years ago
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    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
    • 2 years ago
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    Next time if anyone calls you dumb use the ParthKohli defense!

  59. shubhamsrg
    • 2 years ago
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    ofcorse , as I said, I am gonna show him this screenshot! :)

  60. shubhamsrg
    • 2 years ago
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    Because when he says it, it can't be wrong!

  61. shubhamsrg
    • 2 years ago
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    Not that I am agreeing what he is saying, but he has said it, so no one can argue.

  62. Opcode
    • 2 years ago
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    Hmm. Can I possible get a Ph. D in history? Or is that a no?

  63. shubhamsrg
    • 2 years ago
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    Only one person can answer all the questions in the world! Yes thats right,! We all know who that is. ^_^

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spraguer (Moderator)
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