Sample HTML file: produced with lwarp, with mathematical formulae displayed with MathJax, and xymatrix commutative diagrams displayed with XyJax-v3

6.2 The Galois correspondence for f(t)=t3−2, over the rational field

Let f(t)=t3−2∈Q[t]. Let ω=e2πi3. Let α=23, β=αω and γ=αω2. Hence the set of roots of f is {α,β,γ}. The splitting field of f, over Q is:

Q(α,β,γ)=Q(α,ω).

The monomorphism of groups,

θ:Γ(f,Q)⟶Sym({α,β,γ})τ⟼({α,β,γ}⟶{α,β,γ}a⟼τ(a)), is, in this case, an isomorphism. The diagram of subgroups of Γ(f,Q)≅Sym({α,β,γ}) is below (note that inclusions go in the direction of arrows):

Sym({α,β,γ}){id,(αβγ),(αγβ)}{id,(αβ)}{id,(αγ)}{id,(βγ)}{idR}

The corresponding diagram of intermediate fields Q⊆L⊆Q(α,β,γ) is:

QQ(ω)Q(γ)Q(β)Q(α)Q(α,β,γ)

Also note that {idR}, Sym(R) and {id,(αβγ),(αγβ)}≅A3 are the only normal subgroups of Sym(R). So, respectively,

(4)Q(α,β,γ):Q,Q:Q and Q(ω):Q

are the only normal extensions of Q contained in Q(α,β,γ). Note that the fact that the extensions in (4) are are normal also follows from the fact that they are the splitting fields of p(t)=t3−2, q(t)=t and r(t)=t2+t+1 (which has ω and ω2 as roots), over Q.

  • Remark 3. Also note that we have a series of subfields of C (we use ≤ to denote subfield):

    Q≤Q(ω)≤Q(α,βγ)=Q(α,ω).

    Note that:

    • • Q(ω):Q is a normal extension (since it is the splitting field of t2+t+1, over Q).

    • • Q(ω,α):Q(ω) is also a normal extension. This is because Q(ω,α) is the splitting field of t3−2 over Q(ω).

    And then it follows that:

    • • Γ(Q(ω,α),Q(ω)) is normal in Γ(Q(ω,α):Q),

    • • we have a series of subgroups of Γ(Q(ω,α):Q):

      {e}=Γ(Q(ω,a):Q(ω,α))⊴Γ(Q(ω,α):Q(ω))⊴Γ(Q(ω,α):Q),

    • • The quotient groups can be explicitly determined:

      • – Γ(Q(ω,α):Q)/Γ(Q(ω,a):Q(ω))≅Γ(Q(ω):Q)≅Z2.

        Where the last equation follows since Q(ω):Q is a normal extension of degree 2, since it is the splitting field of the irreducible polynomial t2+t+1, over Q. This Z2 is an abelian group.

      • – Γ(Q(ω,α):Q(ω))/Γ(Q(ω,α):Q(ω,α))≅Γ(Q(ω,α):Q(ω))≅Z3.

        Where the last equation follows since Q(ω,α):Q(ω) is a normal extension of degree 3. This is because it is the splitting field of the polynomial t3−2, which is irreducible over Q(ω). This is an abelian group.

      What was just shown is a general patern that exists any time we compute the splitting field of a polynomial that is soluble by radicals.