MATH 731, FALL 2008
HOMEWORK SET 10 revised
Due Friday, December 12 at 2 p.m.
A. Let Kbe a field. Prove that Kis algebraically closed if and only if it has the fol-
lowing property: Whenever E/F is a finite extension of fields and φ:F→Kis a
homomorphism, there is a homomorphism ψ:E→Ksuch that ψ|F=φ.
B. Let E/F be a finite extension of fields and let K/E/F be a tower of fields. We say K
is a normal closure of E/F if K/F is normal and whenever K/K 0/E/F is a tower of
fields with K0/F normal, we have K0=K. (I.e., there’s no smaller normal extension of
Fbetween Eand K.)
(1) Let E=F(α1, . . . , αn) , let fibe the minimal polynomial of αiover F, and let
f=f1f2·· ·fn. Let Kbe a splitting field of fover E. Prove that Kis a normal closure
of E/F .
(2) Suppose that Kis a normal closure of E /F . Prove that whenever K0/F is a normal
extension and φ:E→K0is an F-homomorphism, there is an F-homomorphism
ψ:K→K0such that ψ|E=φ.
C. Let Fn/Fn−1/Fn−2/···/F1/F0be a tower of fields. You may assume Fn/F0is finite.
Prove that Fn/F0is a separable extension if and only if Fi/Fi−1is a separable extension
for each i= 1, . . . , n .
Do not use Corollary 7.8 in your proof, unless you prove it. (I do not think it is quicker
to prove Corollary 7.8 and use it.)
D. Let f=x4−2∈Q[x] and let E=Q[4
√2, i], so Eis the splitting field for fover Q.
(You don’t need to prove this.) Let G= Gal(f) = Gal(E /Q).
Illustrate the Fundamental Theorem of Galois Theory for E/Q. That is, find the Ga-
lois group Gand find all subgroups of Gand all subfields of E, and show how they
correspond.