What is the splitting field for x^3+1? The splitting field is obtained by adjoining the roots of this polynomial to Q, but I can't get a reasonable answer simply by doing that. Compare with x^3-1, whose splitting field is obtained by adjoining (-1)^(2/3), which gives a degree two extension. Doing similarly in my case creates a splitting field that, to me, seems "too big", namely the splitting field for x^6-1:

The roots of x^3+1 are -1, (-1)^(1/3) and (-1)^(5/3). Adjoining -1 accomplishes nothing, so lets try (-1)^(1/3) =: u. Its minimal polynomial has degree two, so elements of K:=Q(u) are of the form a+bu. Hence u is in K as is its inverse 1-u. Also, all powers of u are in K, thus for example u^2 and u^4 are in K, i.e. the roots of x^3-1 are in K, and consequently the roots of x^6-1 are in K. Hence, spl(x^3+1) = spl(x^6-1).

What am I doing wrong?