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By Scott R.F.

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1. If α ∈ A and λ does not divide α then α3 ≡ ±1 (mod λ4 ). Proof. Since α ≡ 0 (mod λ) then α ≡ ±1 (mod λ). First, we assume α ≡ 1 (mod λ), so α − 1 = βλ where β ∈ A. 5. The Eisenstein Field 43 (α − 1) + (1 − ζ) = βλ + λ = λ(β + 1), α − ζ 2 = (α − ζ) + (ζ − ζ 2 ) = λ(β + 1) + ζλ = λ(β − ζ 2 ). Hence α3 − 1 = (α − 1)(α − ζ)(α − ζ 2 ) = λ3 β(β + 1)(β − ζ 2 ). But 1 − ζ 2 = (1 + ζ)λ, or ζ 2 ≡ 1 (mod λ). Hence β, β+1, β−ζ 2 are in three different classes modulo λ, and at least one is a multiple of λ.

28 (1955), 157–158. , The diophantine equation x3 +y 3 +z 3 +kxyz = 0, Colloque sur la Th´eorie des Nombres, Bruxelles, 1955, Centre Belge Rech. , 1956, pp. 67–76. , Asupra unei teoreme echivalenti cu teorema lui Fermat (On a theorem equivalent to Fermat’s theorem), Gaz. Mat. Fiz. Bucuresti Ser. A, 8 (61) (1956), 23–24; reprinted in Opera Matematicˇ a, Vol. III, pp. 97–98, Edit. Acad. Rep. Soc. Romania, Bucure¸sti, 1973. , Comment on Pedro Piz´ a’s “On the case n = 3 of Fermat’s last theorem,” Math.

342–346. Publi´ees par les soins de MM. Paul Tannery et Charles Henry. Gauthier-Villars, Paris, 1894. , Supplementum quorundam theorematum arithmeticorum quae in non nullis demonstrationibus supponuntur. Novi Comm. Acad. Sci. , 8 (1760/1), 1763, 105– 128. Also in Opera Omnia, Ser. I, Vol. II, pp. 556–575. Teubner, Leipzig, 1915. , Vollst¨ andige Anleitung zur Algebra, 2 volumes. Royal Acad. , St. Petersburg, 1770. English translation by Rev. J. Hewlitt, Longman, Hurst, Rees, Orme, London, 1822.

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