Congruence Classes in Quadratic Integers

In summary, the problem is to find the congruence classes mod (3 + sqrt(-3))/2 in the quadratic field Q[sqrt(-3)]. The ring of integers in this field consists of numbers of the form (a + b*sqrt(-3))/2, where a and b are both even or both odd. The congruence classes are most likely 0, 1, (1 + sqrt(-3))/2, and sqrt(-3) but further testing is needed to confirm this.
  • #1
Frillth
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Homework Statement



I need to find the congruence classes mod (3 + sqrt(-3))/2 in Q[sqrt(-3)].

Homework Equations



None known.

The Attempt at a Solution



I'm not sure how to go about finding these congruence classes. I know that in the regular integers congruences classes mod x are just 0, 1, 2, ... x-1. Do I just try to find all of the quadratic integers with norm less than the norm of (3 + sqrt(-3))/2?
 
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  • #2
I assume you meant in Z[sqrt(-3)]? Or possibly in the ring of integers of Q[sqrt(-3)]?

If nothing else, i think you can just grind out the answer straight from the definition:

[tex]a \equiv b \mod{\frac{3 + \sqrt{-3}}{2}} \Longleftrightarrow
\frac{3 + \sqrt{-3}}{2} \mid (a - b).[/tex]

Or, equivalently, if (a - b) is in the ideal ((3 + sqrt(-3)) / 2).



Actually, it might be easier to compute the ring Z[sqrt(-3)] modulo the ideal ((3 + sqrt(-3)) / 2). It's a relatively straightforward task if:
(1) You've worked with finite fields
(2) You write [itex]\mathbb{Z}[\sqrt{-3}] \cong \mathbb{Z}[x] / (x^2 + 3)[/itex]
(3) You find the smallest (rational) integer in ((3 + sqrt(-3)) / 2).

It might be doable even if you haven't done much with finite fields, and you simply find a single nonzero (rational) integer in that ideal. I think some people work better without using (2), but I think much better if I use (2).

All of these ideas still apply if you are working with the ring of integers of Q(sqrt(-3)), rather than with the ring Z[sqrt(-3)].
 
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  • #3
My specific problem said that I need to work in the quadratic field Q[sqrt(-3)]. I know that this is different than Z[sqrt(-3)], because Z includes only integers of the form a + b*sqrt(-3) where a and b are integers, but Q[sqrt(-3)] includes integers of the from (a + b*sqrt(-3))/2, where a and b are both even or both odd.

I've never heard or integer rings, finite fields, or ideals, and I have no idea what your statement 2 means, so I'm assuming I'm supposed to use a different method to find congruence classes. Any other ideas?
 
  • #4
For every nonzero x in [itex]\mathbb{Q}[\sqrt{-3}][/itex], there is exactly one congruence class modulo x. This is because x is invertible, and so it divides everything.

The set you describe is the ring of integers in [itex]\mathbb{Q}[\sqrt{-3}][/itex]. Equivalently, it is the ring [itex]\mathbb{Z}[ (1 + \sqrt{-3}) / 2][/itex].

The number field [itex]\mathbb{Q}[\sqrt{-3}][/itex] consists of all numbers of the form [itex]a + b \sqrt{-3}[/itex] where a and b are arbitrary rational numbers.


I don't know how well I can help; I really have no idea how this subject is presented if it doesn't assume abstract algebra as a prerequisite. Hopefully someone else can chime in.


Anyways, I think some of what I said is still applicable; you can try and just start at the definitions and grind your way to an answer. And it might help immensely if you can find a rational integer that's in the same congruence class of zero.

What is your definition of congruence class, btw?
 
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  • #5
This is mostly just intuition here, but I'm guessing that the congruence classes I'm looking for are:

0, 1, (1 + sqrt(-3))/2, and sqrt(-3).

Now how do I go about testing whether these are correct?
 

1. What are congruence classes in quadratic integers?

Congruence classes in quadratic integers are sets of numbers that are equivalent to each other under a given modulus. In other words, they are numbers that have the same remainder when divided by the same integer. These classes are an important concept in number theory and have many applications in cryptography and other fields.

2. How are congruence classes in quadratic integers related to modular arithmetic?

Congruence classes in quadratic integers are closely related to modular arithmetic, which is the study of arithmetic operations on integers modulo a given modulus. In modular arithmetic, numbers are grouped into congruence classes based on their remainder when divided by the modulus, and operations such as addition, subtraction, and multiplication are performed on these classes.

3. What is the significance of quadratic integers in congruence classes?

Quadratic integers are numbers of the form a + b√d, where a and b are integers and d is a non-square positive integer. These numbers play a crucial role in congruence classes because they have unique properties that allow for efficient calculations and applications in cryptography, number theory, and other areas of mathematics.

4. How are congruence classes in quadratic integers used in cryptography?

Congruence classes in quadratic integers are used in cryptography to encrypt and decrypt messages. They provide a more secure alternative to traditional modular arithmetic by using quadratic integers, which are more difficult to factorize and therefore make it harder for attackers to break the encryption.

5. Can you give an example of a congruence class in quadratic integers?

One example of a congruence class in quadratic integers is the class of numbers that are congruent to 2 mod 5. This class would include numbers such as 7, 12, and 17, as they all have a remainder of 2 when divided by 5. Another example is the class of numbers that are congruent to 4 mod 9, which would include numbers like 13, 22, and 31.

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