Multiplicative inverse of cosets I wish to find the multiplicative inverse of the coset $(1 + x) + (x^2 + x + 1)Q[x]$ in the field $Q[x]/(x^2 + x + 1)Q[x]$ but am not sure how. Do I start with Euclidean Algorithm?

Note that every coset of $(x^2+x+1) Q[x]$ is of the form $(ax + b) + (x^2+x+1)Q[x]$ by the division algorithm.
The product of two cosets $p + (x^2+x+1)Q[x]$ and $q + (x^2+x+1)Q[x]$ ($p,q \in Q[x]$) is just $pq + (x^2+x+1)Q[x]$.
Therefore, your task is to find $a$ and $b$ such that $(ax + b) \times (x+1)$ leaves a remainder of $1$ when divided by $(x^2+x+1)Q[x]$, so  that the product of those two cosets would  be the identity of the field $Q[x]/(x^2+x+1)Q[x]$.
I will actually hide the solution below : you may see if you are stuck.

 Now, $(ax+b)(x+1) = ax^2 + (a+b)x+ b = (ax^2+ax+a) + (bx + (b-a))$. By uniqueness of the remainder theorem, the remainder is $bx + (b-a)$ which we want equal to $1$. Consequently, we must have $b = 0$ and $b-a = 1$ so $a = -1$. Therefore , the multiplicative inverse is just the element $(-x) + (x^2+x+1)Q[x]$. 

