Fast Computations in the Lattice of Polynomial Rational Function Fields

By Lüroth's Theorem, all intermediate fields of the extension k(x):k, k an arbitrary field, are simple. Those that contain a nonconstant polznomial, the *polynomial rational functions fields*, constitute a sublattice (with respect to set inclusion). We give a fast algorithm for computing a generator of k(p,q), which is similar to the Euclidean algorithm, and also an exteded version, thgat expresses this generator in terms of p and q. These algorithms work over any computable field, in particular, no assumption on the characteristic is needed. Additionally, if k has characteristic 0, we use a deep result of Ritt to give a fast method to compute the other lattice operation, i.e., a generator of the intersection of the fields k(p) and k(q).