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.

Marc Moreno Maza
2008-03-18