# We consider the field E = Q(2, 3, 5). It can be shown that [E : Q]

## Question:

We consider the field E = Q(√2, √3, √5). It can be shown that [E : Q] = 8. In the notation of Theorem 48.3, we have the following conjugation isomorphisms (which are here automorphisms of E):

For shorter notation, let τ_{2} = ψ√_{2.-√2}, τ_{3} = ψ_{√3 -√3}, and , τ_{5} = ψ_{√5.-√5·} Compute the indicated element of E.

Data from Theorem 48.3

Let F be a field, and let α and β be algebraic over F with deg(α, F) = n. The map ψ_{α}_{.β} :F(α) → F(β) defined by ψ_{α}_{.β}(c_{0}+ciα +· · ·+ C_{n-1}α^{n-}¹) = c_{0} + c_{1}β + · · ·+ C_{n-1}β^{n-1} for c_{i }∈ F is an isomorphism of F(α) onto F(β) if and only if a and are conjugate over F.

Proof Suppose that ψ_{α}_{.β} : F(α) → F(β) as defined in the statement of the theorem is an isomorphism. Let irr(σ, F) = a_{0} + a_{1}x + · · · + a_{n}x^{n}. Then a_{0} + a_{1}α + · · · + a_{n}α^{n} = 0, so ψ_{α}_{.β}(a_{0} + a_{1}α + ... + a_{n}α^{n}) = a_{0} + a_{1}β + ... + a_{n}β^{n} = 0.

By the last assertion in the statement of Theorem 29.13 this implies that irr(β, F) divides irr(α, F). A similar argument using the isomorphism (ψ_{α}_{.β})^{-1} = ψ_{β,α} shows that irr(α, F) divides irr(β, F). Therefore, since both polynomials are monic, irr(α, F) = irr(β, F), so α and β are conjugate over F.

Conversely, suppose irr(α, F) = irr(β, F) = p(x). Then the evaluation homomorphisms ∅_{α}: F[x] → F(α) and ∅_{β} : F[x] → F(β) both have the same kernel (p(x)). By Theorem 26.17, corresponding to ∅_{α}: F[x] → F(α), there is a natural isomorphism ψ_{α} mapping F[x]/(p(x)) onto ∅_{α}:[F[x]] = F(α). Similarly, ∅_{β} gives rise to an isomorphism ψ_{β} mapping F[x]/(p(x)) onto F(β). Let ψ_{α}_{.β} = ψ_{β}(ψ_{α})^{-1}. These mappings are diagrammed in Fig. 48.4 where the dashed lines indicate corresponding elements under the mappings. As the composition of two isomorphisms,ψ_{α}_{.β} is again an isomorphism and maps F(α) onto F(β). For (c_{0} + c_{1}α + · · + c_{n-1}a^{n-1}) ∈ F(α), we have

Thus ψ_{α,}_{β} is the map defined in the statement of the theorem.

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