Exercise 15.4.13

Proof. We know (see p. 498) that, for an odd Gaussian integer $\beta \in \mathbb{Z}[i]$,

where

We have proved in Exercise 7 that $P_{\beta },Q_{\beta }$ are in $\mathbb{Z}[i][u]$, thus the numbers $a_i$ are Gaussian integers.

The numbers $b_k(\beta )$ are defined for $k\in \mathbb{N}$ by

where the right member is a formal series of $ℂ[[u]]$, and $u$ a variable.

Starting from the sum of geometric series $1∕(1+x)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{k=0}^{\infty }{(-1)}^k{x}^k$, we obtain

where the last sum makes sense in the ring $ℂ[[u]]$ of formal series, since

so a given power of $u$ appears in only finitely many terms.

Therefore, writing $a_i=a_i(\beta )$, and $a_0=1$

Developing ${(a_1+\cdots +a_d{u}^{d-1})}^k$ with the Multinomial Theorem, we obtain

whose coefficients are Gaussian integers. Therefore the developing of the right member is a formal series with only Gaussian integer coefficients, and $b_k(\beta )\in \mathbb{Z}[i]$ for all indices $k\ge 0$.

Since $i^{𝜀}\frac{u^d+a_1(\beta )u^{d-1}+\cdots +a_d(\beta )}{1+a_1(\beta )u+\cdots +a_d(\beta )u^d}$ is analytic in the neighborhood of $0$, the radius of convergence of this series is nonzero. □