Showing the Galois group of Q(⁴√2, i) over Q is isomorphic to D4

[~Death~]

E=Q(4th root of 2, i) and G is the galios group of E over Q

I found the minimal polynomial p(x) of 4th root of 2 over Q and Q(i) to be
x^4-2

I'm trying to show

(1) the galios group H of E over Q(i) is a normal subgroup of G

(2) If K is the galios group of Q(i) over Q show that it is isomorphic to G/H

so I can ultimately show that G is actually D4 (the group of symmetries)

but I'm compeltely stuck

 

[HallsofIvy]

Okay, what have you done so far? What are the roots of the polynomial $x^4= 2$? What is G? What is H?

By the way- it is 'Galois theory'. Capital G because it is a person's name and o before i.

 

[~Death~]

Okay, what have you done so far? What are the roots of the polynomial $x^4= 2$? What is G? What is H?

I found the minimal polynomial of 4th root of 2 over Q and Q(i) to be
x^4-2

and the roots are +/-w, +/-wi where w is the 4th root of 2

 

[Petek]

Additional hint: What is the splitting field of $x^4 - 2$ over Q?

Petek