ASKSAGE: Sage Q&A Forum - Individual question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 29 Jun 2014 10:05:47 -0500Compute Galois closure of an extension of a function fieldhttp://ask.sagemath.org/question/7875/compute-galois-closure-of-an-extension-of-a-function-field/Say I want to look at the field extension $Quot(\mathbb{Q}[x,y]/y^7-x)$ over $\mathbb{Q}(x)$ and then compute its Galois closure. How do I do that?
Ideally it could be done on the scheme-level (to define the scheme-morphism: (the projectivization of the affine plane curve $y^7-x$) mapping to (the projective $x$-line); and then compute its Galois closure -- a scheme!). But I don't know how to implement either version.Sat, 15 Jan 2011 14:11:48 -0600http://ask.sagemath.org/question/7875/compute-galois-closure-of-an-extension-of-a-function-field/Comment by Evgeny for <p>Say I want to look at the field extension $Quot(\mathbb{Q}[x,y]/y^7-x)$ over $\mathbb{Q}(x)$ and then compute its Galois closure. How do I do that?</p>
<p>Ideally it could be done on the scheme-level (to define the scheme-morphism: (the projectivization of the affine plane curve $y^7-x$) mapping to (the projective $x$-line); and then compute its Galois closure -- a scheme!). But I don't know how to implement either version.</p>
http://ask.sagemath.org/question/7875/compute-galois-closure-of-an-extension-of-a-function-field/?comment=22210#post-id-22210i think this question needs a better title. "how to implement this computation?" does not tell anything about the content.Sun, 30 Jan 2011 20:50:57 -0600http://ask.sagemath.org/question/7875/compute-galois-closure-of-an-extension-of-a-function-field/?comment=22210#post-id-22210Answer by vdelecroix for <p>Say I want to look at the field extension $Quot(\mathbb{Q}[x,y]/y^7-x)$ over $\mathbb{Q}(x)$ and then compute its Galois closure. How do I do that?</p>
<p>Ideally it could be done on the scheme-level (to define the scheme-morphism: (the projectivization of the affine plane curve $y^7-x$) mapping to (the projective $x$-line); and then compute its Galois closure -- a scheme!). But I don't know how to implement either version.</p>
http://ask.sagemath.org/question/7875/compute-galois-closure-of-an-extension-of-a-function-field/?answer=23124#post-id-23124Hi,
Currently you can create such a field
sage: K.<x> = FunctionField(QQ, 'x')
sage: R.<y> = PolynomialRing(K)
sage: p = y^7 - x
sage: K2 = K.extension(p)
sage: K2
Function field in y defined by y^7 - x
But, sadly, there is no such feature as computing the Galois closure...
VincentSun, 29 Jun 2014 10:05:47 -0500http://ask.sagemath.org/question/7875/compute-galois-closure-of-an-extension-of-a-function-field/?answer=23124#post-id-23124