Is the (regular) inverse Galois problem known for the field C(x,y)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:20:12Z http://mathoverflow.net/feeds/question/27118 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27118/is-the-regular-inverse-galois-problem-known-for-the-field-cx-y Is the (regular) inverse Galois problem known for the field C(x,y)? Makhalan Duff 2010-06-04T23:32:56Z 2010-06-05T01:59:13Z <p>I'd be surprised if somebody proved the inverse Galois problem for $\mathbb{C}(x,y)$, but I wanted to make sure.</p> http://mathoverflow.net/questions/27118/is-the-regular-inverse-galois-problem-known-for-the-field-cx-y/27123#27123 Answer by James Borger for Is the (regular) inverse Galois problem known for the field C(x,y)? James Borger 2010-06-05T01:20:16Z 2010-06-05T01:59:13Z <p>Surely the inverse Galois problem is known over $\mathbf{C}(x)$: The Galois group of the maximal extension of $\mathbf{C}(x)$ unramified away from $n+1$ given primes of $\mathbf{C}[x]$ is the free profinite group on $n$ generators. Any finite group $G$ is a quotient of such a group, so there exists a finite Galois extension $L/\mathbf{C}(x)$ with Galois group $G$. </p> <p>Then $L\otimes_{\mathbf{C}(x)}\mathbf{C}(x,y)$ is a finite Galois extension of $\mathbf{C}(x,y)$ with Galois group $G$.</p>