Exercise 1.

Let $a = x^8 + 1$ , $b = x^4 + x + 1$ , $f = x^4 + x^3 + 1$ be in ${\mbox{${\mathbb{Q}}$}}[x]$ .

1. Compute the inverse of $f$ modulo $x^5$ .
2. Deduce the quotient and the remainder in the Euclidean division of $a$ w.r.t. $b$ .

Note that the example has been chosen such that computations can easily be carried out by hand.

Answer 1  

Answer 2