So I have this polynomial with coefficients on a Galois Field GF(256):
f(x) = (00011011)*x^4 + (00011011)*x^3 + (10110001)*x^2 + (10110000)*x + (00000001)
I wrote a program to solve it (by trying every one of the 256 possible roots), but I only found two roots:
00000001
11110100
I'm not familiar with abstract algebra and I'm not a mathematician, I was expecting 4 distinct roots since it is modelling 4 errors on a message. Now I'm not sure if it is my program that has some bug and failed to find the other roots (I think that this is unlikely since it only fails once every 2500 trials), or if I just misunderstood something about how abstract algebra works. A degree 4 polynomial should have 4 roots even if its coefficients are elements of a group, right?
f(x) = (00011011)*x^4 + (00011011)*x^3 + (10110001)*x^2 + (10110000)*x + (00000001)
I wrote a program to solve it (by trying every one of the 256 possible roots), but I only found two roots:
00000001
11110100
I'm not familiar with abstract algebra and I'm not a mathematician, I was expecting 4 distinct roots since it is modelling 4 errors on a message. Now I'm not sure if it is my program that has some bug and failed to find the other roots (I think that this is unlikely since it only fails once every 2500 trials), or if I just misunderstood something about how abstract algebra works. A degree 4 polynomial should have 4 roots even if its coefficients are elements of a group, right?
