Prove that the polynomial p(x)=x4−4x2+8x+2 is irreducible over the quadratic field F =Q(−2). [Hint : first use the method of proposition 11 for the U.F.D Z[−2](cf.Exercise 8, Section 8.1) to show that if α∈Z[−2] is a root of p(x) then α is a divisor of 2 in Z[−2] . Conclude that αα must be ±1,±−2 or ±2 and hence show that p(x) has no linear factor overFF. Show similarly that p(x) is not the product of quadratics with coefficients in F.]
