TITLE: Algorithm generalizing continued fractions for non-quadratic algebraic numbers
QUESTION [17 upvotes]: The continued fraction algorithm generates an integer sequence which terminates for a rational number, is periodic for the roots of irreducible integer quadratics, and is non-periodic for other algebraic numbers.  This sequence uniquely determines the number in a useful way, e.g. one can compute convergents and solve Diophantine equations.
Does there exist a corresponding algorithm for, say, roots of irreducible cubics which has similar properties?  What about other algebraic numbers?  What is known about this?  Or to save people time, what phrase should I google to find out the answer?

REPLY [6 votes]: Suppose that r is a positive real number with no nontrivial positive real Galois conjugates.  This case includes the case of nth roots of integers.  Then it is not difficult to work out the continued fraction of r algorithmically.
Suppose that f(x) is the minimal polynomial of r.  The greatest integer of r (which is a_0) is just the largest integer n such that f(n) is negative.  Then it's easy to work out the minimal polynomial of 1/(r-a_0) using the fact that the minimal polynomial of the reciprocal is given by reversing the coefficients.  Furthermore, 1/(r-a_0) again has the same property as r (no nontrivial positive real Galois conjugates).  Now compute a_1 as the largest integer such that f(a_1) is negative.  Wash, rinse, repeat.