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Galois group of a quartic over Q
Posted:
Dec 27, 2007 10:06 PM
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Greetings,
Problem: Let f(x) = x^4 + a*x^2 + b, be a polynomial with coefficient
in Q, the rational numbers. Assume f(x) is irreducible over Q (and no
multiple roots). I want to find the Galois group of it over Q.
It is very easy to show that it is a subgroup of D_4 the Dihedral-4
group (of order 8). And indeed I it is not hard to see that if b is a
square in Q, then the Galois group must be (Z/2Z)^2, the Klein-4
group. And with a little more effort I can show that if b(a^2 - 4*b)
is a square in Q, then the Galois group must be Z/4Z. Finally, I think
if neither b nor b(a^2 - 4*b) is a square in Q, then the Galois group
must be D_4, and that's where I got stuck.
Attempt: We know the roots are {u,-u, v,-v}. Then (u^2)(v^2)=b. Since
f(x) is irreducible, [Q(u) : Q]=4. If I can show that v is not in
Q(u), then I am all set, since that forces [Q(u,v) : Q]=8, and all
subgroups of S_4 of order 8 are isomorphic to D_4. But how can show
this?
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