Abstract Algebra, Euclidean Algorithm

[kuahji]

Use the Euclidean Algorithm to find the gcd of the given polynomials:

(x³-ix²+4x-4i)/(x²+1) in C[x]

First I got x-i R: 3x-3i, then I took the 3x-3i into x²+1 & got 1/3 x R: 1+i. Then I was going to take 1+i into 3x-3i. However that never ends it seems, unless I just confused myself.

The answer in the back of the book is just x-i for the gcd. So I'm thinking I should have stopped earlier, & believe I'm just over looking something simple. Any suggestions?

 

[mighty2000]

Hello there,

First of all, great job using the Euclidean Algorithm to find the gcd of these polynomials! You are definitely on the right track, but it seems like you may have made a small mistake along the way.

Let's go through the steps together to see where things may have gone wrong:

1. Divide x^3 - ix^2 + 4x - 4i by x^2 + 1:
(x^3 - ix^2 + 4x - 4i) / (x^2 + 1) = x - i

2. Now, we need to find the remainder when (x^2 + 1) is divided by (x - i). This is where you got stuck, but it's important to remember that we are working in the complex numbers (C[x]), so we can have complex numbers as coefficients in our polynomials.

(x^2 + 1) / (x - i) = (x - i)(x + i) / (x - i) = x + i

3. Now, we can continue with the Euclidean Algorithm:
(x - i) / (x + i) = 1/3 (x - i) + 2/3 (1 + i)

4. Finally, we divide (x + i) by 1/3 (x - i) + 2/3 (1 + i):
(1 + i) / (1/3 (x - i) + 2/3 (1 + i)) = 1/3 (1 + i)

Therefore, the gcd of (x^3 - ix^2 + 4x - 4i) and (x^2 + 1) in C[x] is 1/3 (1 + i). However, we can simplify this further to just x - i, which is the same as the answer in the back of the book.

I hope this helps clarify things for you. Keep up the good work!