Find all f(x) and g(x) in Z_3[x]: (x^3+x+1)f(x)+(x^2+x+1)g(x)=1

Aron Heath

Aron Heath

Answered question

2022-11-18

Need help understanding finite fields / modulo for polynomials
I'm taking a class in finite fields and have not been able to conceptualize how modulo + finite fields works in polynomial space. I understand the basic premises of modular arithmetic, but can't work out how to actually generate a finite field of polynomials.
For example:
Find all f(x) and g(x) in Z 3 [ x ]:
( x 3 + x + 1 ) f ( x ) + ( x 2 + x + 1 ) g ( x ) = 1
I know conceptually how to solve this sort of equation when the coefficients are integers and f(x),g(x) are simple variables, but I don't know how to generate fields in Z 3 [ x ] and then how exactly to use them to solve this sort of equation for polynomials once I have their gcd in Z 3 [ x ].

Answer & Explanation

Ricardo Weiss

Ricardo Weiss

Beginner2022-11-19Added 12 answers

Let the the consecutive numbers be :
(x) and (x+1)
As per the condition given:
(x)+(x+1)=9
2x+1=9
2x=8
x=4
x=4
So the numbers are as follows: 4,5

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