Exercise 3.26

Use Ex. 25 to show that if ξ , η , ζ ∈ ℤ [ ω ] are not zero and ξ 3 + η 3 + ζ 3 = 0 , then λ divides at least one of the elements ξ , η , ζ .

Answers

Proof. Let ξ , η , ζ ∈ ℤ [ ω ] ∖ { 0 } such that ξ 3 + η 3 + ζ 3 = 0 .

Reasoning by contradiction, suppose that λ ∤ ξ , λ ∤ η , λ ∤ ζ .

By Ex. 3.24,

ξ ≡ ± 1 ( mod λ ) , η ≡ ± 1 ( mod λ ) , ζ ≡ ± 1 ( mod λ ) ,

and by Ex.3.25,

ξ 3 ≡ ± 1 ( mod 9 ) , η 3 ≡ ± 1 ( mod 9 ) , ζ 3 ≡ ± 1 ( mod 9 ) ,

As ± 1 ± 1 ± 1 ≢ 0 ( mod 9 ) , this is a contradiction.

Conclusion : if ξ , η , ζ are not zero and ξ 3 + η 3 + ζ 3 = 0 , then λ divides at least one of the elements ξ , η , ζ .

(Consequence : if x 3 + y 3 + z 3 = 0 , x , y , z ∈ ℤ , then 3 ∣ xyz : this is the first case of Fermat’s theorem for the exponent 3.) □

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2022-07-19 00:00
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