Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

LolWolf Group Title

Here's a little brain teaser, that I feel some would enjoy: Given some Gaussian Integer \(z\in \mathbb{Z}[i]\) find and prove a closed-form expression for the number of equivalence classes in \(\mathbb{Z}[i]/z\mathbb{Z}[i]\)

  • one year ago
  • one year ago

  • This Question is Closed
  1. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    It might seem easy to show, but it's not as easy as you might think to prove. I will say, though, there *is* a pretty neat proof.

    • one year ago
  2. KingGeorge Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Could you give me a brief explanation of what \(\mathbb{Z}[i]/z\mathbb{Z}[i]\) means? Specifically what it means to do the mod operation.

    • one year ago
  3. KingGeorge Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    \[\mathbb{Z}[i]=\{a+bi\;|\;a,b\in\mathbb{Z}\} \quad\text{and}\quad i=\sqrt{-1}\]

    • one year ago
  4. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Yeah, sorry, I was out. But, here: So, we have, like @KingGeorge said, \[ \mathbb{Z}[i]=\{a+bi\;|\;a,b\in \mathbb{Z}\}\text{ where }i^2=-1 \]While we have:\[ \mathbb{Z}[i]/z\mathbb{Z}[i]=\{n\in\mathbb{Z}[i]\;|\forall m\in\mathbb{Z}[i], n\equiv m\;(\!\!\!\!\!\!\mod z)\} \]

    • one year ago
  5. KingGeorge Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    So in the complex numbers, modular arithmetic takes place as follows correct?\[a+bi\pmod{c+di}\equiv\left[ a\mod{c}+bi\mod{d}\right]\]

    • one year ago
  6. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    I don't quite understand your case... But in the Gaussian Integers, the modulus has the same definition as the normal integers: \[ a\equiv b\mod c \iff c\,|\,(b-a)\\ a, b, c \in \mathbb{Z}[i] \]

    • one year ago
  7. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Wait, why'd my previous reply not make it? I forgot to add to the previous other post that the norm of the integer must be less than that of the modulus.

    • one year ago
  8. KingGeorge Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    That's what I thought, but just wanted to verify.

    • one year ago
  9. KingGeorge Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    I must go, but I will think about this.

    • one year ago
  10. Herp_Derp Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    So your definition isn't quite correct; it should be:\[\large\mathbb{Z}[i]/z\mathbb{Z}[i]=\{n\in\mathbb{Z}i:\forall m\in\mathbb{Z}[i]~~\text{with}~~|m|<|z|,~n\equiv m\pmod{z}\}\]

    • one year ago
  11. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Yes, it's not, it'd be easier to show using division algorithm... but, frankly, I'm too lazy to re-type it.

    • one year ago
  12. Herp_Derp Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    And I'm too anally autistic to not correct it :)

    • one year ago
  13. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Haha, I'll re-type it later, I guess...

    • one year ago
  14. KingGeorge Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Just fyi for people reading, how I thought the modular arithmetic worked at first, was not correct.

    • one year ago
  15. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    Here's the better definition: \[ \mathbb{Z}[i]/z\mathbb{Z}[i]=\{r\in\mathbb{Z}[i]\;|\;\forall n\in\mathbb{Z}[i], q\in\mathbb{Z}[i], r=n-qz, \,N(r)<\frac{1}{2}N(z)\} \]

    • one year ago
  16. LolWolf Group Title
    Best Response
    You've already chosen the best response.
    Medals 0

    (Quintessentially, you have to remember that these are the sets of a single element (Gaussian Integer), which represents an equivalence class)

    • one year ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.