# Zariski problem on field extensions

The Zariski problem has its motivation in a geometric question. For example, one could ask the following: Given two curves $C_1$ and $C_2$, make two surfaces by crossing a line with each curve. If the resulting surfaces are isomorphic, must the original curves also be isomorphic?
In general, one starts with two affine varieties, $V_1$ and $V_2$, of dimension $n$ (cf. also Affine variety) and crosses each with a line. Associated to each $V_i$ is its coordinate ring $c[V_i]$, and from an algebraic point of view, one wants to know if the polynomial rings $c[V_1][x_1]$ and $c[V_2][x_2]$ being isomorphic forces the coordinate rings to be isomorphic (cf. also Isomorphism). For $n$ larger than two, this is an open problem (as of 2000). However, also associated to each $V_i$ is its function field, $c(V_i)$, and one wants to know if isomorphism of the rational function fields in one variable over the function fields forces the function fields to be isomorphic. This is the so-called Zariski problem.
The problem has an affirmative answer for varieties of dimension one. This result appears in [a3], but uses ideas from [a5] and in an essential way depends on Amitsur's results about function fields of genus zero [a1]. Using a wide range of ideas from algebraic geometry, [a2] provides a family of counterexamples to the problem. In particular, there exist a field $K$ and extension fields $L$ of transcendence degree two over $K$ that are not rational and yet $L(x_1,x_2,x_3)$ is a pure transcendental extension of $K$ in five variables. Finally, in [a4] it is shown that the problem does have an affirmative answer most of the time, i.e., if the original varieties are of general type. Again, this result uses [a6] and, in an essential way, the results from [a7].