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Compute Galois closure of an extension of a function field

asked 2011-01-15 14:11:48 -0600

Oliver gravatar image

updated 2015-01-13 11:15:53 -0600

FrédéricC gravatar image

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.

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i think this question needs a better title. "how to implement this computation?" does not tell anything about the content.

Evgeny gravatar imageEvgeny ( 2011-01-30 20:50:57 -0600 )edit

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answered 2014-06-29 10:05:47 -0600

Hi,

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...

Vincent

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Asked: 2011-01-15 14:11:48 -0600

Seen: 198 times

Last updated: Jun 29 '14