Compute the G.C.D of two Gaussian Integers

In summary, the conversation is about finding the Greatest Common Divisor (G.C.D) of two Gaussian integers, a=14+2i and b=21+26i. The norm of a and b are found to be 200 and 1117, respectively. It is suggested that any common divisor must also divide the G.C.D of the norms. Since 1117 and 200 are co-prime, their G.C.D is 1, making the G.C.D of a and b a unit (1, -1, i, -i) in the ring. It is then discussed that in the integers, any factor that divides (a+bi) must also divide (a+bi)(a-bi
  • #1
DeldotB
117
8

Homework Statement


Hello all I apologize for the triviality of this:
Im new to this stuff (its easy but unfamiliar) I was wondering if someone could verify this:

Find the G.C.D of [itex]a= 14+2i [/itex] and [itex]b=21+26i [/itex].

[itex] a,b \in \mathbb{Z} [ i ] [/itex] - Gaussian Integers

Homework Equations



None

The Attempt at a Solution



Well, is it true that any common divisor must also divide the G.C.D of the norm's of [itex] a [/itex]and[itex] b [/itex]?

If so then, [itex] norm(14+2i)=200 [/itex]
[itex]norm(21+26i)=1117 [/itex]

Well, since 1117 and 200 are co-prime, their greatest common divisor is one. Thus,

Thus the G.C.D of a,b is a unit (1,-1,i,-i) in the ring.

Thanks
 
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  • #2
Everything that divides (a+bi) also divides (a+bi)(a-bi), sure.
DeldotB said:
Well, since 1117 and 200 are co-prime
In the integers. You'll have to show that this is true for Gaussian integer factors as well.
 

FAQ: Compute the G.C.D of two Gaussian Integers

1. What are Gaussian Integers?

Gaussian Integers are complex numbers that can be expressed in the form of a + bi, where a and b are both integers and i is the imaginary unit (√-1).

2. How do I compute the G.C.D of two Gaussian Integers?

To compute the G.C.D of two Gaussian Integers, you can use the Euclidean algorithm. This involves repeatedly finding the remainder when one number is divided by the other until the remainder is 0. The last non-zero remainder is the G.C.D.

3. Can the G.C.D of two Gaussian Integers be negative?

Yes, the G.C.D of two Gaussian Integers can be negative. This is because the G.C.D is defined as the greatest common divisor, not necessarily the greatest common positive divisor.

4. How is the G.C.D of two Gaussian Integers used in mathematics?

The G.C.D of two Gaussian Integers is used in many mathematical concepts, such as finding the least common multiple, solving Diophantine equations, and factoring polynomials. It is also a fundamental concept in number theory.

5. Can I use a calculator to compute the G.C.D of two Gaussian Integers?

While some calculators may have a function for computing the G.C.D of two integers, most calculators do not have a function for Gaussian Integers. It is best to use a computer program or manually calculate the G.C.D using the Euclidean algorithm.

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