Let α=32∈R, and let ω be a primitive cube root of unity in C, so ω3=1. (i) Prove that the set of all numbers p+qα+rα2, for p,q,r∈Q, is a subfield of C. Hint: You can consider ω as 2−1+3i. Also note that {1,α,α2},{1,ωα,(ωα)2}, and {1,ω2α,ωα2} are linearly independent over Q. Also, in computing the inverse of p+qα+rα2, you can use the identity

Solution As in the question given, alpha=root(3)(2)inRR and let omega be a primitive cubed root of unity in CC, so omega^3=1.

Resuelto Let α=32∈R, and let ω be a primitive cube root of

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Pregunta: Let α=32∈R, and let ω be a primitive cube root of unity in C, so ω3=1. (i) Prove that the set of all numbers p+qα+rα2, for p,q,r∈Q, is a subfield of C. Hint: You can consider ω as 2−1+3i. Also note that {1,α,α2},{1,ωα,(ωα)2}, and {1,ω2α,ωα2} are linearly independent over Q. Also, in computing the inverse of p+qα+rα2, you can use the identity

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Solución
Paso 1
Solution
As in the question given,
$α = \sqrt[3]{2} \in R$ and let $ω$ be a primitive cubed root of unity in $C,$ so $ω^{3} = 1 .$
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Paso 2
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Texto de la transcripción de la imagen:

Let

α = 32 \in R

, and let

ω

be a primitive cube root of unity in

C

, so

ω^{3} = 1

. (i) Prove that the set of all numbers

p + q α + r α^{2}

, for

p, q, r \in Q

, is a subfield of

C

. Hint: You can consider

ω

\frac{- 1 + 3 i}{2}

. Also note that

{1, α, α^{2}}, {1, ω α, (ω α)^{2}}

, and

{1, ω^{2} α, ω α^{2}}

are linearly independent over

Q

. Also, in computing the inverse of

p + q α + r α^{2}

, you can use the identity

(p + α + r^{2}) (p + q α + r^{2} α^{2}) (p + q^{2} α + r α^{2}) = p^{3} + 2 (q^{2} - 3 p r) + ω^{2}

(ii) Show that the map

p + q α + r α^{2} \mapsto p + q ω α + r ω^{2} α^{2}

is a monomorphism onto its image, but not an automorphism.

(p + q α + r α^{2}) (p + q ω α + r ω^{2} α^{2}) (p + q ω^{2} α + r ω α^{2}) = p^{3} + 2 (q^{3} - 3 pq r) + 4 r^{3}