Finding the inverse of a polynomial in a field

[Adorno]

Homework Statement

(A somewhat similar question to my last one). Let $J$ be the ideal of the polynomial ring $\mathbb{Q}[x]$ generated by $x^2 + x + 3$. Find the multiplicative inverse of $(3x^3 + 3x^2 + 2x -1) + J$ in $\mathbb{Q}[x]/J$

Homework Equations

The Attempt at a Solution

I think I need to apply the extended Euclidean algorithm to 3x^3 + 3x^2 + 2x -1 and x^2 + x + 3 in order to find the greatest common divisor, but I am unsure of the details. Also, once I find the gcd, I don't know what I'm supposed to do with it.

 

[christoff]

You are indeed correct that you must use the extended Euclidean algorithm with those two polynomials. If you are unfamiliar with the general (or extended) Euclidean algorithm, then I first recommend reviewing this subject. The algorithm is exactly the same for polynomials as it is for integers, so use the same strategy here and you should be good for that part.

Then of course, the question is, what to do once you have the gcd? Let's write F=3x^3+3x^2+2x-1 and J=x^2+x+3 for notational convenience. Since J is irreducible and monic, by definition of the gcd, gcd(F,J) must be either H or some nonzero rational number. As you might guess, it's a rational number (find it!). Say gcd(F,J)=q.

The extended Euclidean algorithm then yields an expression for q as a linear combination of F and J, that is, q=pF+gJ for some polynomials p and g. q is a nonzero rational, so
$$q q^{-1}=1=q^{-1}(pF+gJ).$$
But then in the factor ring Q[x]/J, the additive unit is 0=0+J where (boldface) J is the ideal generated by the polynomial J. Work with the above expression, and I think you will have enough to answer the question :)