A cubic ring of integers with the smallest Pythagoras number (Q2071772)

The author proves that the ring of integers in the unique number field \(K^{(49)}= \mathbb{Q}(\zeta_7+ \zeta^{-1}_7)\) of discriminant \(49\) has Pythagoras number equal to \(4\). Moreover, it is determined which numbers are sums of integral squares in this field. An application is presented in the construction of a diagonal universal quadratic form in \(5\) variables, namely \(x^2_1+ x^2_2+ x^2_3+ x^2_4+ (1+\rho+\rho^2)x^2_5\), where \(\varphi= \zeta_7+ \zeta^{-1}_7\).