PROBLEM SET # 11
MATH 114
Due April 20.
1. Let Eand Bbe normal extensions of Fand E∩B=F. Prove that
AutFEB ∼
=AutFE×AutFB.
2. Find the Galois group of the polynomial (x3−3) (x3−2) over Q.
3. Let f(x) be an irreducible polynomial of degree 7 solvable in radicals. List all
possible Galois groups for f(x).
4. Find the Galois group of x6−4x3+ 1 over Q.
5. Let f(x) = g(x)h(x) be a product of two irreducible polynomials over a finite
field Fq. Let mbe the degree of g(x) and nbe the degree of h(x). Show that the
degree of the splitting field of f(x) over Fqis equal to the least common multiple of
mand n.
6. Let F⊂Ebe a normal extension with Galois group isomorphic to Z2×···×Z2.
Assuming that char F6= 2, prove that
E=Fpb1,...,pbs
for some b1,...,bs∈F.
7. Prove that the splitting field of the polynomial x4+ 3x2+ 1 over Qis isomorphic
to Qi, √5.
Date: April 12, 2006.
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