※ 引述《PttFund (批踢踢基金只進不出)》之銘言：
: Let F be a field and let f(x) be an irreducible polynomial
: of degree 3 over F. Suppose that L is an extension of F of
: degree 6 such that f(x) splits into linear factors over L.
: Prove that there exists a proper intermediate field K such
: that K is Galois over F.
Suppose f(x) is separable.
1. If L is the splitting over F of f(x),
then the Galois group of f(x) over F is isomorphic to S3
which has a normal subgroup A3.
Hence the fixed field of A3 is Galois over F.
2. Otherwise the splitting field K of f(x) over F
is a proper intermediate field which is Galois over F.
Suppos f(x) is inseparable.(This may happen)
Then there exists a intermediate field K such that [K:F]=2
Hence K is Galois over F.
ex:
F=Z3(t^3) {The rational fucntion field over Z3}
f(x)= x^3 -t^3 is irreducible over F
with splitting field Z3(t) over F which is not Galois over F.
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好像寫太長了 本來也沒有這麼長的
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