Let k ∈ ℤ be such that ${|}_k\left(\mathbb{Q}\right)|$ is finite, where ${}_k:y²=1-2kx+k²x²-4x³$. We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.

Let ε be a quartic algebraic unit. We give necessary and sufficient conditions for (i) the quartic number field K = ℚ(ε) to contain an imaginary quadratic subfield, and (ii) for the ring of algebraic integers of K to be equal to ℤ[ε]. We also prove that the class number of such K's goes to infinity effectively with the discriminant of K.

Nous déterminons sous certaines hypothèses, un système fondamental d’unités du corps non pur $K=\mathbb{Q}\left(\omega \right)$ et de son sous-corps quadratique, où $\omega$ est solution du polynôme$$f\left(X\right)=X^4+d^{-2}{M}_6{X}^2-M_4,$$avec $M_6=D^6+6D^{4}d+9D^2{d}^2+2d^3$, $M_4=D^4+4D^{2}d+2d^2$, $d|D$, $d$, $D\in \mathbb{N}$, non nuls.