View source for Zariski problem on field extensions

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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|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|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 [[#References|[a3]]], but uses ideas from [[#References|[a5]]] and in an essential way depends on Amitsur's results about function fields of genus zero [[#References|[a1]]]. Using a wide range of ideas from [[Algebraic geometry|algebraic geometry]], [[#References|[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 [[#References|[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 [[#References|[a6]]] and, in an essential way, the results from [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Amitsur, "Generic splitting fields for central simple algebras" ''Ann. of Math.'' , '''2''' : 62 (1955) pp. 8–43 {{MR|70624}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Beauville, J.-L. Colliot-Thelene, J.-J. Sansuc, P. Swinnerton-Dyer, "Varietes stablement rationnelles non rationnelles" ''Ann. of Math.'' , '''121''' (1985) pp. 283–318 {{MR|0786350}} {{ZBL|0589.14042}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Deveney, "Ruled function fields" ''Proc. Amer. Math. Soc.'' , '''86''' (1982) pp. 213–215 {{MR|0667276}} {{ZBL|0529.12016}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Deveney, "The cancellation problem for function fields" ''Proc. Amer. Math. Soc.'' , '''103''' (1988) pp. 363–364 {{MR|0943046}} {{ZBL|0652.12013}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Nagata, "A theorem on valuation rings and its applications" ''Nagoya Math. J.'' , '''29''' (1967) pp. 85–91 {{MR|0207688}} {{ZBL|0146.26302}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J. Ohm, "The ruled residue theorem for simple transcendental extensions of valued fields" ''Proc. Amer. Math. Soc.'' , '''89''' (1983) pp. 16–18 {{MR|0706500}} {{ZBL|0523.12021}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Roquette, "Isomorphisms of generic splitting fields of simple algebras" ''J. Reine Angew. Math.'' , '''214/5''' (1964) pp. 207–226 {{MR|0166215}} {{ZBL|0219.16023}} </TD></TR></table>