Splitting Fields?

Find the splitting field of x^4+1 over Q.Splitting Fields?
The splitting field by definition is the smallest field that has all the roots of the polynomial.





So the roots of x^4+1==0 are...





(sqrt(2)+i*sqrt(2))/2,


(-sqrt(2)+i*sqrt(2))/2,


(-sqrt(2)-i*sqrt(2))/2, and


(sqrt(2)-i*sqrt(2))/2





Call the first one a. Since the other three are a^3, a^5, and a^7 respectively, the field we are looking for is


Q((sqrt(2)+i*sqrt(2))/2).





Since x^4+1 is the minimum polynomial (4th degree) to which (sqrt(2)+i*sqrt(2))/2 is a root. The field we have is any number in the form of (4 terms):





w + x* (sqrt(2)+i*sqrt(2))/2)+y * i + z*(sqrt(2)-i*sqrt(2))/2)





w,x,y, and z are in Q.





Note that the x coefficient is a, the one for y is a^2, and z a^3.