In the paper
http://annals.math.princeton.edu/2007/165-2/p04
Theorem 2. Let $b \ge 2$ be an integer. The b-ary expansion of any irrational
algebraic number cannot be generated by a finite automaton. In other
words, irrational automatic numbers are transcendental.
And in the following paper http://adamczewski.perso.math.cnrs.fr/ABK_JA.pdf
Theorem 1.1 (Christol et al.). Let $p_1$ and $p_2$ be distinct prime numbers and let $q_1$ and $q_2$ be
powers of prime $p_1$ and $p_2$, respectively. Let $(a_n)_{n\ge0}$ be a sequence with values in a finite set
$\mathcal{A}$ with cardinality at most $min(q_1, q_2)$. Let $i_1$ and $i_2$ be two injections from $\mathcal{A}$ into $\mathbb{F}_{q_1} $and $\mathbb{F}_{q_2}$ ,
respectively. Then, the formal power series
$$f (t) =\sum_{n\ge 0}i_1(a_n)t^n \in \mathbb{F}_{q_1}((t))$$ and $$ g(t) =\sum_{n\ge 0}i_2(a_n)t^n \in \mathbb{F}_{q_2}((t))$$
are both algebraic (respectively over $\mathbb{F}_{q_1}((t))\text{ and }\mathbb{F}_{q_2}((t))$ if and only if they are rational functions.
They are seemingly contradict to each other over algebraicity and transcendency. I am kind of puzzled about this.
Could anyone explain the difference between the two theorems, especially by examples?