For $\beta > 1$ a real algebraic integer ({\it the base}), the finite alphabets $\mathcal{A} \subset \mathbb{Z}$ which realize the identity $\mathbb{Q}(\beta) = {\rm Per}_{\mathcal{A}}(\beta)$, where ${\rm Per}_{\mathcal{A}}(\beta)$ is the set of complex numbers which are $(\beta, \mathcal{A})$-eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base $\beta$ and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in $\mathbb{Q}(\beta)$, generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases $\gamma_s := \gamma_{n, m_1 , \ldots , m_s}$ such that $\gamma_{s}^{-1}$ is the unique root in $(0,1)$ of an almost Newman polynomial of the type $-1+x+x^n +x^{m_1}+\ldots+ x^{m_s}$, $n \geq 3$, $s \geq 1$, $m_1 - n \geq n-1$, $m_{q+1}-m_q \geq n-1$ for all $q \geq 1$. For $\beta > 1$ a reciprocal algebraic integer close to one, the poles of modulus $< 1$ of the dynamical zeta function of the $\beta$-shift $\zeta_{\beta}(z)$ are shown, under some assumptions, to be zeroes of the minimal polynomial of $\beta$.

翻译：对于 $\ beta > 1美元 一个真正的正方位整值( 基数 ), 调查固定的字母 $\ mathcal{A}\ subset\ mathb} 美元, 使身份 $\ mathb{A\\ (\ beta) = perm Per\ mathcal{A\\ (\ beta) 美元, $\ 美元 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 美元, 固定的字母, 与贪婪的算法相匹配。 3qeta 美元 和 Lehmer 问题。 开始重写线索的概念是构建与基础小多元值相联的中间字母 。 以 $\ = * 美元, 美元为源子的表示 美元, 美元, 美元, 美元 美元, 普通的 Schemiax 序列 和 数字有关 。