How Do I Calculate The Galois Group And Intermediate Fields Of A Certain Extension Using Magma?

Asked Oct 18 '23 at 14:53

Active Oct 18 '23 at 17:32

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here MAGMA Commands for Galois Theory calculations it is discussed how to calculate the galois group when the the field that is fixed is say the rationals. But what if the field that is fixed is $\Bbb{F}_2(x)$ the polynomial ring with coefficients in $\Bbb{F}_2$.

To be specific i look at the normal closure of extension $\Bbb{F}_2(x) \subset \Bbb{F}_2(x,y,w)$ where $y$ and $w$ satisfy equations:

$$ y^2 + y = x^3 + x,\qquad (x^7 + x + 1)(w^2 + w) = (x^5 + x) y + x^2 + x $$

This extension of the normal closure is finite and galois but how does one obtain the galois group + the intermediate fields as i tried using commands Subfields() and GaloisGroup() but these only work i believe if the ground field is the rationals or some number field. Thoughts?

edited Oct 18 '23 at 17:32

Jyrki Lahtonen

140,891

asked Oct 18 '23 at 14:53

Harry Crane

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For some basic information about writing mathematics at this site see, e.g., here, here, here and here. – Another User Oct 18 '23 at 14:56
Both $\Bbb{F}_2(x,y)/\Bbb{F}_2(x)$ and $\Bbb{F}_2(x,y,w)/\Bbb{F}_2(x,y)$ are Artin-Schreier extensions (hence cyclic Galois of degree two). But I don't see why $\Bbb{F}_2(x,y,w)/\Bbb{F}_2(x)$ should be Galois? The intermediate field has the non-trivial automorphism $y\mapsto y+1$, but I don't see if that extends to the bigger field. – Jyrki Lahtonen Oct 18 '23 at 17:38
What normal closure exactly are you referring to? – Jyrki Lahtonen Oct 18 '23 at 17:41
F2(x)⊂F2(x,y,w) – Harry Crane Oct 25 '23 at 16:13

How Do I Calculate The Galois Group And Intermediate Fields Of A Certain Extension Using Magma?

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