Let where . Verify that is also a root of . Determined . Show that is normal over .

means . So we only need to prove is another root of .

Let , means .

Let , then , obviously , thus is a root of .

is monic and has no root in , so it is irreducible in . is minimal polynomial of . So . must be the cyclic group of order 3. By theorem 4.7 chapter 4 in Jacoboson’s book, is normal over .

Jacobson, Algebra I. p243.

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