5 Let F Z5 Is there a polynomial gx such that gx2x1 is cong

5. Let F = Z₅. Is there a polynomial g(x) such that g(x)(2x+1) is congruent to 1 modulo x²+1? If yes, please find g, if not, please prove it.

Solution

let g(x) = ax + b modulo (x^2 + 1)

we want (ax+b)(2x + 1) = 1 modulo (x^2+1) in Z5

i.e.

2ax^2 + 2bx + ax + b = 1 modulo ( x^2 + 1) in z5

i.e.

(subtracting 2ax^2 + 2a)

2bx + ax + b - 2a = 1 modulo (x^2+1) in z5

So (2b +a) = 0 in z5

and (b-2a) = 2 in Z5

from 1 ) a = -2b mod 5

putting in 2

-5b = 2 mod5

Not possible at all

So no solution