Apart from Wiles'proof (which I didn't read), I know 4 proofs of the Fermat-Wiles theorem for exponent 3, i. e. impossibilty of $x^{3} + y^{3} + z^{3} = 0$ with x, y and z pairwise coprime nonzero rational integers :

1° Euler's proof (decomposition in the 3d cyclotomic field, which is a quadratic field, leads to an infinite descent); proves that there are no nontrivial solutions in rational numbers;

2° Kummer's proof; also a decomposition in the 3d cyclotomic field and an infinite descent, but the result is stronger than Euler's one : the equation has no nontrivial solution in the 3d cyclotomic field;

3° decomposition in $\mathbb{Q}(\sqrt[3]{2})$ : squaring $x^{3} + y^{3} = - z^{3}$ and substracting $4 x^{3} y^{3}$, we find that $z^{6} - 4 x^{3} y^{3}$ is a square in $\mathbb{Z}$. In view of arithmetic properties of the field $\mathbb{Q}(\sqrt[3]{2})$, this implies that $z^{2} - xy (\sqrt[3]{2})^{2}$ is a square in $\mathbb{Z}[\sqrt[3]{2}]$. This leads to an infinite descent;

4° decomposition in $\mathbb{Q}(\sqrt[3]{3})$ : we have for example $x + y = a^{3}, y + z = 9b^{3}, x + z = c^{3}$ with a, b, and c rational integers. This implies $a^{3} + 9b^{3} + c^{3} = 6 abc$.

(See for example Legendre, Second supplément, p. 7, section 9, online : https://books.google.fr/books?id=zHMs_R2SATYC&printsec=frontcover&hl=fr#v=onepage&q&f=false )

This can be written $(2c + 3b \sqrt[3]{3} + a (\sqrt[3]{3})^{2}) (9ab - 4c^{2} + (6bc - 3a^{2})\sqrt[3]{3} + (2ac - 9b^{2})(\sqrt[3]{3})^{2}) = c^{3}$

and also

$N_{\mathbb{Q}}^{\mathbb{Q}(\sqrt[3]{3})}(2c + 3b \sqrt[3]{3} + a (\sqrt[3]{3})^{2}) = - c^{3}$

and in view of arithmetic properties of $\mathbb{Q}(\sqrt[3]{3})$, this implies that there exist rational integers r, s, t such that

$2c + 3b \sqrt[3]{3} + a (\sqrt[3]{3})^{2} = (-2 + (\sqrt[3]{3})^{2}) (r + s \sqrt[3]{3} + t (\sqrt[3]{3})^{2})^{3}$

and $N_{\mathbb{Q}}^{\mathbb{Q}(\sqrt[3]{3})}(r + s \sqrt[3]{3} + t (\sqrt[3]{3})^{2}) = -c.$

This leads to an infinite descent. (I know, this fourth proof is ugly, but we are here for the fun.) The third proof was already published, but the fourth was perhaps not (I found it myself).

My question is : do you know other proofs of the Fermat-Wiles theorem for exponent 3 ? In particular, can anybody say if the paper of Z. Xu : "Another proof of Fermat's Last Theorem for Cubes", J. Southwest Teach. Univ., Ser. B (1987), No 1, 20-22, which claims to use no complex numbers, is accurate and different from proofs 3 and 4 above ? (This paper is in Chinese...)