Function fields of conics as invariant subfields (Q1320187)

The authors have written an exposition for non-specialists concerning function fields of conics as invariant subfields. The following is the authors' summary. Section 1 contains some basic definitions. In Section 2 a rationality criterion for function fields of quadratic forms is proved; this is a central result which includes the observation that the function field of a quadratic \(k\)-form is rational iff the form has a simple \(k\)-zero. The basic properties of function fields of conics and, in particular, their interpretation as fields of invariants are discussed in Section 3. Section 4 is concerned with the rationality of function fields of surfaces fibered by conics, i.e., of function fields of polynomials of the form \(X^ 2 - cY^ 2 - f(U)\). As an application of previous results, necessary and sufficient conditions are given in order for the function field of such a polynomial to be rational when the base field \(k\) is the real field and the degree of \(f(U)\) is \(\leq 4\). The proof of the rationality theorem of Section 2 yields an algorithm for finding an algebraically independent generating set for a rational function field of a quadratic form; this is illustrated with an example in Section 5. It is sometimes possible in treating quadratic forms in 4 variables to pass to an appropriate quadratic extension of \(k\), thereby reducing the problem to one in 3 variables, i.e., to the case of a conic. We apply this technique to the rationality problem in Section 6.