Urgent need please help THANK YOUSolutionWe have ax x4 2x2

Urgent need!! please help!! THANK YOU.

Solution

We have a(x)= x⁴ 2x² + 9. Let x² = y. Then the given expression changes to a(x)= y² -2y + 9 , where y = x² . The roots of the polynomial are y =[-(-2) ± { ( -2)² – 4*1*9}]/ 2*1 = [2 ±(4 – 36)]/2 = (2 ±-32)/2 =( 2 ± 4i 2)/2 = 1± 2 i 2 Thus, the roots of a(x) are x = ( 1+2i 2)^1/2, -(1+2i 2)^1/2, (1- 2 i 2 )^1/2 and - (1- 2 i 2 )^1/2 . Since the roots of a(x) are linear combinations of 2 and I, we say that Q (2, i ) is the root field of a(x) over Q.