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Rational numbers have finite continued fractions, and quadratic algebraic numbers over $\mathbb Q$ have eventually periodic continued fraction representations.

Is there a way to recognise a different kind of algebraic number (e.g. third-degree algebraic numbers?) from its continued fraction representation? It seems to arbitrary that we only have such a nice characteristic for numbers of degree 2 over $\mathbb Q$ in $\mathbb R$.

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    $\begingroup$ This is called Hermite's problem. One version of the problem is dynamical in nature, and slightly stronger than the one I suspect you're asking; this one has been answered negatively. As for the one you're likely asking, check the Wikipedia page. $\endgroup$ – AJY Mar 2 '17 at 16:22
  • $\begingroup$ @AJY It doesn't seem (from the Wikipedia page) to be exactly the problem I'm asking, though it is definitely interesting. I'd like to see if there is anything else than periodicity that you can check of a continued fraction, from which one can conclude that the number belongs to e.g. the cubic irrationals or the quartic irrationals. $\endgroup$ – tomsmeding Mar 2 '17 at 16:33
  • $\begingroup$ as far as I know there is no such way to recognize algebraic numbers; recommend the inexpensive cambridge.org/us/academic/subjects/mathematics/number-theory/… $\endgroup$ – Will Jagy Mar 2 '17 at 17:35
  • $\begingroup$ Ah, my apologies. As for that, my guess is no. $\endgroup$ – AJY Mar 2 '17 at 18:03
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    $\begingroup$ This article might interest you, but it uses Bissinger's generalization of continued fractions. $\endgroup$ – ccorn Mar 3 '17 at 1:08

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