Symmetric functions, Galois groups, and intermediate fields - sci.math

Message from discussion Symmetric functions, Galois groups, and intermediate fields

dave.rusin

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More options Nov 20 2009, 6:01 pm

Newsgroups: sci.math

From: "dave.rusin" <dave.ru...@gmail.com>

Date: Fri, 20 Nov 2009 10:01:20 -0800 (PST)

Local: Fri, Nov 20 2009 6:01 pm

Subject: Re: Symmetric functions, Galois groups, and intermediate fields

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On Nov 19, 11:10 pm, Kusanagi <Kusan...@hotmail.com> asked about
identifying intermediate fields in the Galois correspondence,
in a specific case.

Recap: inside the function field F(x,y,z) we identify the subfield
F(r,s,t) of symmetric functions, where r=x+y+z, s=xy+yz+zx, t=xyz.
Then F(x,y,z) is a Galois extension of F(r,s,t) with galois group
S_3.

> Let D = (x-y)(y-z)(x-z) [...]
> let X=D^2 and consider E=F(r,s,t)[ X^(1/2) ].
> Once we square D, it seems like every permutation is fixed.

You mean: every permutation in S_3 fixes X=D^2. That's correct.

> Is it true that Gal(E:F(r,s,t)) = S_3?

No, you jumped the gun. You need the small field to be fixed
(element-wise) by the group, yes. But you also need the large field
to be fixed setwise by the group (it is), and you need to know that
everything not in the small field is moved by some element of the
group (it is), and you need every element of the group to move
_something_ in the big field (not true here).

Indeed, look at the definition of E: it's a quadratic extension of
F(r,s,t), hence a Galois extension with a Galois group of order 2.

Your mistake was to observe only that F(r,s,t) is fixed by all
of S_3, so that there is a homomorphism of S_3 _into_ the
Galois group you seek; but that homomorphism has a kernel, because
all of the alternating subgroup of S_3 fixes not only F(r,s,t)
but the larger field E as well.

> Is it possible to find an intermediate field E explicitly such
> that Gal(E/F(r,s,t)) = Z_2, Z_3, and A_3 (alternating group of
> degree 3), where E is a subfield of F(x,y,z)?

Um, what is "Z_3" if not A_3 ?

I guess you know about the Galois correspondence: the subgroups
of S_3 are in one-to-one correspondence with the intermediate
fields between F(x,y,z) and F(r,s,t). In this particular case
we can describe the correspondence explicitly. Here is a table
showing the different subgroups H of S_3 and the corresponding
subfields of F(x,y,z) :

H = {1} corresponds to F(x,y,z) itself
H = (12) corresponds to F( x+y, xy, z)
H = (13) corresponds to F( x+z, xz, y)
H = (23) corresponds to F( y+z, yz, x)
H = (123) corresponds to E = F( r,s,t, D )
H = S_3 corresponds to F( r, s, t)

You might want to check all the features of the Galois
correspondence: that H1 < H2 implies E1 > E2; that H1=normal
iff E / F(r,s,t) is Galois; and most importantly that in
each case the subfield consists precisely of all the elements
of F(x,y,z) that are invariant under the action of H.

dave

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