sage: R.<X> = PolynomialRing(QQ)
sage: K.<a> = NumberField( X^4 - 2*X^2 + 2 )
sage: L.<s> = K.galois_closure()
sage: K.is_galois()
False
sage: L.degree()
8
sage: RK.<Y> = PolynomialRing(K)
sage: RK
Univariate Polynomial Ring in Y over Number Field in a with defining polynomial X^4 - 2*X^2 + 2
sage: RL.<Z> = PolynomialRing(L)
sage: RL
Univariate Polynomial Ring in Z over Number Field in s with defining polynomial X^8 - 20*X^6 + 104*X^4 - 40*X^2 + 1156
sage: a.minpoly()
x^4 - 2*x^2 + 2
sage: s.minpoly()
x^8 - 20*x^6 + 104*x^4 - 40*x^2 + 1156
sage: factor( Y^2 - 2 )
Y^2 - 2
sage: # so sqrt(2) is not in K
sage: factor( Z^2 - 2 )
(Z - 1/328*s^6 + 15/328*s^4 + 3/82*s^2 - 155/164) * (Z + 1/328*s^6 - 15/328*s^4 - 3/82*s^2 + 155/164)
sage: # so sqrt(2) is in L
sage: G.<g> = L.galois_group()
sage: G.structure_description()
'D4'