(a) Find the minimal polynomial ( f(x) ) of ( alpha=sqrt{8+sqrt{15}} ) over ( Q ) and prove that your answer is correct. (b) Find the Galois group of the splitting field of ( f(x) ) over ( Q ).
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This is a polynomial in \( Q[x] \) that has \( \alpha \) as a root. We need to check that it is irreducible and that it is the minimal polynomial of \( \alpha \). To see that it is irreducible, we can use the Eisenstein criterion with the prime \( p = 7 \). Show more…
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