Exercise 3, § 1

Commutative Algebra, IX, p. 42

a) There exists a unique sequence of polynomials $(R_n)_{n \in \mathbf{N}}$ in $\mathbf{Z}[X, Y]$ such that, for every integer $n \geqslant 0$ , one has

X^{p^n} + Y^{p^n} = \sum_{i=0}^n p^i R_i(X, Y)^{p^{n-i}}.

b) If $p$ is odd, one has, for every $n \in \mathbf{N}$ ,

R_n(X, -X) = 0.

If $p = 2$ , one has

R_1(X, Y) = -XY

and

R_n(X, X) \equiv 0 \mod 2 \quad \text{for } n \geqslant 2.

Solution

No solution has been written yet.