In my class notes, I have two theorems which don't quite seem to fit together. Maybe you can help me out.(adsbygoogle = window.adsbygoogle || []).push({});

Thm 1If p(x) in F[x] splits in K, then E=F(a_{1},...,a_{n}) is the splitting field of p(x) in K (the a_i's are the roots of p(x)).

Thm 2If p(x) in F[x], then the splitting field of p(x) is unique up to isomorphism.

I'm clearly missing something big here. Doesn't (1) imply (2)? Isn't (1) even stronger than (2)?

What's an example of a polynomial with two distinct but isomorphic splitting fields?

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# Uniqueness of Splitting Fields

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