View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300601.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300602.png" />, 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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300604.png" />, of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300605.png" /> (cf. also [[Affine variety|Affine variety]]) and crosses each with a line. Associated to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300606.png" /> is its coordinate ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300607.png" />, and from an algebraic point of view, one wants to know if the polynomial rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z1300609.png" /> being isomorphic forces the coordinate rings to be isomorphic (cf. also [[Isomorphism|Isomorphism]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006010.png" /> larger than two, this is an open problem (as of 2000). However, also associated to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006011.png" /> is its function field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006012.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006013.png" /> and extension fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006014.png" /> of transcendence degree two over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006015.png" /> that are not rational and yet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006016.png" /> is a pure transcendental extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130060/z13006017.png" /> 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>