Show by an example that Corollary 50.6 is no longer true if the word irreducible is deleted.
Question:
Show by an example that Corollary 50.6 is no longer true if the word irreducible is deleted.
Data from Corollary 50.6
Corollary
If E
< F̅ is a splitting field over F, then every irreducible polynomial in F[x] having a zero in E splits in E.
Proof If E is a splitting field over F in F̅, then every automorphism of F̅ induces an automorphism of E. The second half of the proof of Theorem 50.3 showed precisely that E is
also the splitting field over F of the set {gk(x)} of all irreducible polynomials in F[x]
having a zero in E. Thus an irreducible polynomial f(x) of F[x] having a zero in E has
all its zeros in F̅ in E. Therefore, its factorization into linear factors in F̅[x], given by
Theorem 31.15, actually takes place in E[x], so f(x) splits in E.
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