(a) Let f(x) = x4 − x2 − 6 and suppose α ∈ C is a root of f(x). Is f(x) irreducible over

Question

(a) Let f(x) = x4 − x2 − 6 and suppose α ∈ C is a root of f(x). Is f(x) irreducible over

Q? What is [Q[α] : Q]? As always, justify your answer.

(b) Let g(x) = x7 − 3x4 − 6x + 15 and suppose β ∈ C is a root of g(x). Is g(x) irreducible

over Q? What is [Q[β] : Q]?

please answer as if answering an upper level abreast algebra college exam question

Answer 1

a) (x^2-3)(x^2+2)=0

x=+/-sqr3, +/-isqr2

4 roots, 2 imaginary