How to interpret quotient rings of gaussian integers

In summary, the conversation discusses the elements of Z[i]/I, where I is an ideal generated by a non-zero non-unit integer. It is found that the elements are isomorphic to the integers mod 3 adjoin i and the quotient ring I is the (integers adjoin i) mod the generator of the ideal.
  • #1
nateHI
146
4

Homework Statement



This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on.

What are some of the elements of [itex]Z/I[/itex] where I is an ideal generated by a non-zero non-unit integer. For the sake of argument, let's take I=<3>.

Homework Equations


The Attempt at a Solution


Representatives from one coset would be the following...
(8+4i)/<3>=(5+i)/<3>=(2+i)[itex]\in Z/<3>[/itex]

Representatives from another coset would be the following...
(7+5i)/<3>=(4+2i)/<3>=(1+2i)[itex]\in Z/<3>[/itex]
 
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  • #2
At first glance I think it would be [itex]\mathbb{Z}_3[/itex]. Can you try to prove this?
 
  • #3
http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/Zinotes.pdf

I found the answer in that link on page 17. You are correct the elements are isomorphic to the integers mod 3 adjoin i. I just didn't understand that the quotient ring I was trying to produce the elements of was the (integers adjoin i) mod the generator of the ideal.
 

1. What are Gaussian integers?

Gaussian integers are complex numbers of the form a + bi, where a and b are integers and i is the imaginary unit (i.e. the square root of -1). They are named after the mathematician Carl Friedrich Gauss.

2. How do you interpret quotient rings of Gaussian integers?

To interpret a quotient ring of Gaussian integers, you can think of it as dividing the set of Gaussian integers into equivalence classes, where two Gaussian integers are equivalent if their difference is a multiple of a given Gaussian integer (known as the modulus). The resulting quotient ring is a new mathematical structure that can help solve certain problems involving Gaussian integers.

3. What is the significance of interpreting quotient rings of Gaussian integers?

Interpreting quotient rings of Gaussian integers allows for a deeper understanding of the structure and properties of Gaussian integers. It also has practical applications in number theory, cryptography, and signal processing.

4. How do you compute a quotient ring of Gaussian integers?

To compute a quotient ring of Gaussian integers, you must first choose a Gaussian integer to be the modulus. Then, you divide the set of Gaussian integers by the modulus, keeping only the remainders. The resulting set of remainders forms the quotient ring.

5. What are some real-world examples of using quotient rings of Gaussian integers?

Quotient rings of Gaussian integers are used in various fields, such as error-correcting codes in telecommunication, encryption algorithms in cryptography, and digital signal processing in audio and image compression. They are also used in the study of elliptic curves, which have applications in physics and computer science.

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