subfields of the splitting field $K=\mathbb{Q}(\sqrt[3]{3},\sqrt{2},\sqrt{3}i)$.

Question

I proved that $K=\mathbb{Q}(\sqrt[3]{3},\sqrt{2},\sqrt{3}i)$ is a splitting field of the polynomial $(x^3 -3)(x^2 -2)$ whose galois group is isomorph to $S_3 \times \mathbb{Z}_2 $ and this group have a one element of order 1, 7 elements of order 2, 1 element of order 3, 3 elements of order 4, 3 elements of order 6 and 1 element of order 12. Hence i have the subfields of $K$ by the by galois correspondence are

order 1 is $K$

order 3 is $\mathbb{Q}(\sqrt[3]{3})$

order 6 are $\mathbb{Q}(\sqrt[3]{3},\sqrt2)$, $\mathbb{Q}(\sqrt[3]{3},\sqrt{3}i)$, $\mathbb{Q}(\sqrt[3]{3},\sqrt{6}i)$.

order 4 are $\mathbb{Q}(\sqrt{2},\sqrt{3}i)$, $\mathbb{Q}(\sqrt{2},\sqrt{6}i)$,$\mathbb{Q}(\sqrt{3}i,\sqrt{6}i)$.

order 12 is $\mathbb{Q}$

and I do not know which are the 7 fields of degree 4. am I doing well? thanks.

Answer 1

Let us check the situation by asking the engine...

sage: R.<x> = PolynomialRing(QQ)
sage: f = (x^3 - 3) * (x^2 - 2)
sage: K.<a> = f.splitting_field()
sage: K.defining_polynomial()
x^12 + 6*x^10 - 12*x^9 + 24*x^8 + 110*x^6 + 504*x^5 - 66*x^4 + 420*x^3 + 1284*x^2 + 36*x + 1
sage: a.minpoly()
x^12 + 6*x^10 - 12*x^9 + 24*x^8 + 110*x^6 + 504*x^5 - 66*x^4 + 420*x^3 + 1284*x^2 + 36*x + 1
sage: K.degree()
12
sage: Gal = K.galois_group()
sage: Gal.structure_description()
'D6'
sage: Gal.order()
12

sage: K.is_galois()
True

... and we can also ask for the subfields...

sage: for subfield_data in K.subfields():
....:     L = subfield_data[0]
....:     print(f'DEGREE {L.degree()} :: defining polynomial = {L.defining_polynomial()}')

DEGREE 1 :: defining polynomial = x
DEGREE 2 :: defining polynomial = x^2 + 12*x + 144
DEGREE 2 :: defining polynomial = x^2 - 18
DEGREE 2 :: defining polynomial = x^2 + 54
DEGREE 3 :: defining polynomial = x^3 + 24
DEGREE 3 :: defining polynomial = x^3 - 192
DEGREE 3 :: defining polynomial = x^3 + 24
DEGREE 4 :: defining polynomial = x^4 + 18*x^2 + 324
DEGREE 6 :: defining polynomial = x^6 + 12*x^5 + 96*x^4 + 304*x^3 + 672*x^2 + 6528*x + 18496
DEGREE 6 :: defining polynomial = x^6 - 6*x^4 + 6*x^3 + 12*x^2 + 36*x + 1
DEGREE 6 :: defining polynomial = x^6 + 18*x^4 + 6*x^3 + 108*x^2 - 108*x + 225
DEGREE 6 :: defining polynomial = x^6 - 6*x^4 - 48*x^3 + 12*x^2 - 288*x + 568
DEGREE 6 :: defining polynomial = x^6 + 18*x^4 - 48*x^3 + 108*x^2 + 864*x + 792
DEGREE 6 :: defining polynomial = x^6 + 18*x^4 + 6*x^3 + 108*x^2 - 108*x + 225
DEGREE 6 :: defining polynomial = x^6 - 6*x^4 + 6*x^3 + 12*x^2 + 36*x + 1
DEGREE 12 :: defining polynomial = x^12 + 6*x^10 - 12*x^9 + 24*x^8 + 110*x^6 + 504*x^5 - 66*x^4 + 420*x^3 + 1284*x^2 + 36*x + 1

So the splitting field of the given polynomial $f=(x^3-3)(x^2-2)$ is a field $K$ of degree $12$ over $\Bbb Q$. It is Galois over $\Bbb Q$. The Galois field has order $12$, and its structure is D6, the dihedral group with $12$ elements.

sage: Gal.is_isomorphic(DihedralGroup(6))
True

The subfields above correspond to the subgroups of the $D_6$ (sometimes denoted with sub index $12$, reflecting the order but...)

sage: D6 = DihedralGroup(6)
sage: subgroups = D6.subgroups()
sage: subgroups.sort(key=lambda H: 12/H.order())
sage: for H in subgroups:
....:     print(f'ORDER {H.order()} :: Normal? {H.is_normal()} :: Commutative? {H.is_commutative()}')
....: 
....: 
ORDER 12 :: Normal? True :: Commutative? False
ORDER 6 :: Normal? True :: Commutative? False
ORDER 6 :: Normal? True :: Commutative? True
ORDER 6 :: Normal? True :: Commutative? False
ORDER 4 :: Normal? False :: Commutative? True
ORDER 4 :: Normal? False :: Commutative? True
ORDER 4 :: Normal? False :: Commutative? True
ORDER 3 :: Normal? True :: Commutative? True
ORDER 2 :: Normal? False :: Commutative? True
ORDER 2 :: Normal? False :: Commutative? True
ORDER 2 :: Normal? False :: Commutative? True
ORDER 2 :: Normal? True :: Commutative? True
ORDER 2 :: Normal? False :: Commutative? True
ORDER 2 :: Normal? False :: Commutative? True
ORDER 2 :: Normal? False :: Commutative? True
ORDER 1 :: Normal? True :: Commutative? True

(Although the most information comes here in code and from code, i hope that the structure is clear enough to put explicitly the hands on the objects... Of course, sage can give more, if we really need more...)