Questions

$(1)$

Consider $a = x^2 + x + 1$ and $b = x - 1$ . Using the Euclidean division, compute a $b$ -adic expansion of $a$

$(2)$

For the general case, design an algorithm that computes a $b$ -adic expansion of $a$ .

$(3)$

Give an upper bound for the time complexity of this algorithm, counting the number of operations in ${\mbox{${\mathbb{K}}$}}$ .

$(4)$

Show that $a$ admits one and only one $b$ -adic expansion.