Let $\Sigma$ be an alphabet of size $t$, let $f:\Sigma^*\rightarrow\Sigma^*$ be a non-erasing
morphism, let $\mathbf w$ be an infinite fixed point of $f$, and let $E(\mathbf w)$ be the critical
exponent of $\mathbf w$. We prove that if $E(\mathbf w)$ is finite, then for a uniform $f$ it is
rational, and for a non-uniform $f$ it lies in the field extension
$\Q[\lambda_1,\ldots,\lambda_\ell]$, where $\lambda_1,\ldots,\lambda_\ell$ are the distinct
eigenvalues of the incidence matrix of $f$. In particular, $E(\mathbf w)$ is algebraic of degree at
most $t$. We also give an algorithm for computing $E(\mathbf w)$.
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{\bf MSC}: 68R15\\
{\bf Key Words}: Critical exponent; Circular D0L languages
