6.3 Splitting Fields
1.
a. x3 + 1 = (x + 1)(x2 − x + 1), and the roots of x2 − x + 1 are 
and 
so 
But 
so it follows that 
Since 
is a root of the polynomial x2 + 3, and since x3 + 3 is irreducible over 
(no root), then it is the minimal polynomial of 
Hence 
and so But so it follows that Since is a root of the polynomial x2 + 3, and since x3 + 3 is irreducible over (no root), then it is the minimal polynomial of Hence
c. f = (x2 − 7)(x2 + 1) so 
. Thus,
. Thus,
Moreover, because x2 − 2 is irreducible over and because x2 + 1 has no root in Hence by the multiplication ...
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