Finding inverse in polynomial factor ring

[PsychonautQQ]

Homework Statement

find the inverse of r in R = F[x]/<h>.
r = 1 + t - t^2
F = Z_7 (integers modulo 7), h = x^3 + x^2 -1

Homework Equations

None

The Attempt at a Solution

The polynomial on bottom is of degree 3, so R will look like:
R = {a + bt + ct^2 | a,b,c are elements of z_7 and x^3 = 1 - ^2}

To solve this problem I realized that the inverse must obviously have the form of some element in R, so I set up:
(a + bt + ct^2)(1 + t - t^2) = 1

then I multiplied it all out whilst continuously substituting for t^3 and then solving for coefficients where the constant coefficient should equal 1 and the other two should equal 0.

I did all of this and got the constant coefficient to be zero and nonzero answers for the other two >.<. I checked my calculations and can't find an error (doesn't necessarily mean there isn't one...), is something wrong with the way I set up the problem? is my substitution for x^3 correct?

 

[Dick]

Homework Statement

find the inverse of r in R = F[x]/<h>.
r = 1 + t - t^2
F = Z_7 (integers modulo 7), h = x^3 + x^2 -1

Homework Equations

None

The Attempt at a Solution

The polynomial on bottom is of degree 3, so R will look like:
R = {a + bt + ct^2 | a,b,c are elements of z_7 and x^3 = 1 - ^2}

To solve this problem I realized that the inverse must obviously have the form of some element in R, so I set up:
(a + bt + ct^2)(1 + t - t^2) = 1

then I multiplied it all out whilst continuously substituting for t^3 and then solving for coefficients where the constant coefficient should equal 1 and the other two should equal 0.

I did all of this and got the constant coefficient to be zero and nonzero answers for the other two >.<. I checked my calculations and can't find an error (doesn't necessarily mean there isn't one...), is something wrong with the way I set up the problem? is my substitution for x^3 correct?

The usual way to do this is to use the Extended Euclidean Algorithm to explicitly write Bezout's identity. gcd(h,r)=a*h+b*r. Then mod both sides by h. Does that sound familiar? It is kind of a tedious calculation and it's easy to make a mistake. What did you do with t^4?