Decompose number in Gaussian interger field

• tsang
In summary, To decompose a number into irreducible factors inside a Gaussian integer field, you can use the fact that every Gaussian integer has an integer norm, and that the norm is multiplicative. You can first factor the number's norm and then look for Gaussian integers with that norm. For example, to factor 13, you can factor its norm 169 and then look for Gaussian integers with norm 13. Once you have found the Gaussian integers, you can check if they can be factored further by checking if their norms are prime. If the norms are prime, then the Gaussian integers are irreducible and cannot be factored further.
tsang
Hi everyone, I always have trouble on decomposing number into irreducible factors inside Gaussian integer field. I keep trying to express number as product as (a+bi)(c+di), and trying to solve a,b,c,d inside of integers (Z), then see if they are irreducibles, which of course end of very messy.

Is there any techniques or algorithms to do these kind of decomposing numbers? Can anyone show me an example please so I may get good idea? Say take a prime 5, try to decompose it into irreducibles in Z

Thanks a lot.

You should use that every gaussian integer has an integer norm, and that the norm is multiplicative. The norm is defined as N(a+bi)=a2+b2.

So if you like to factor the number 13, you first factor its norm 169=13*13, and then look for gaussian integers of norm 13=9+4, so you only need to consider ±3±2i, and ±2±3i, and indeed 13=(3+2i)(3-2i), and this can not be factored further since the norms of the factors are prime.

Hi Norwegian, thank you so much for your reply. It is very clear, thank you.

I have another question now, that for example 13=(3+2i)(3-2i), but since my goal is decompose 13 into irreducibles, how do I show (3+2i) and (3-2i) are irreducibles, so they cannot be decomposed further? Do I have to repeat the whole process like how I decomposed 13 to try to decompose (3+2i) and (3-2i) again?

Thanks a lot for your time.

As I told you above, 3+2i can not be factored further, since the norm N(3+2i)=13 is a prime in Z, (and any factor of 3+2i would have a norm dividing 13). We don't count the units (norm=1) ±1,±i as factors btw.

There are several techniques and algorithms that can be used to decompose numbers in Gaussian integer fields. One common method is to use the Euclidean algorithm, which involves repeatedly dividing the given number by a Gaussian integer and finding the greatest common divisor. This process can be continued until the resulting Gaussian integer is irreducible.

Another approach is to use the concept of norms in Gaussian integer fields. The norm of a Gaussian integer is defined as the product of the number and its complex conjugate. By finding the norms of different Gaussian integers that divide the given number, one can determine the irreducible factors.

To demonstrate this, let's take the prime number 5 and decompose it in the Gaussian integer field Z. The norm of 5 is 25. Now, we can find the Gaussian integers that divide 25 - these are 1, 5, and 1+2i. Out of these, only 1+2i is a Gaussian prime, as its norm is 5. Therefore, we can decompose 5 as (1+2i)(1-2i).

In general, the process of decomposing numbers in Gaussian integer fields can be complex and may require knowledge of advanced mathematical concepts. It is always helpful to seek guidance from a mathematician or refer to reliable sources for assistance.

1. What is a Gaussian integer field?

A Gaussian integer field is a mathematical concept in which complex numbers are represented as a combination of real and imaginary parts, where the real and imaginary parts are both integers. It is denoted as ℤ[i] and is a subset of the complex numbers.

2. How do you decompose a number in a Gaussian integer field?

To decompose a number in a Gaussian integer field, you need to factorize the number into its prime factors. Then, you can express the number as a product of Gaussian integers, where each Gaussian integer is raised to a power based on its corresponding prime factor.

3. Why is decomposing numbers in a Gaussian integer field important?

Decomposing numbers in a Gaussian integer field is important because it allows for easier and more efficient calculations with complex numbers. It also helps in solving mathematical problems involving complex numbers, such as finding roots of polynomials.

4. Can any number be decomposed in a Gaussian integer field?

No, not all numbers can be decomposed in a Gaussian integer field. Only integers that can be expressed as a sum of two squares can be decomposed in a Gaussian integer field.

5. What is the significance of using a Gaussian integer field in number theory?

Gaussian integer fields are significant in number theory because they provide a unique way of representing complex numbers and allow for easier calculations and factorization of complex numbers. They also have applications in cryptography and algebraic geometry.

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