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We prove that the ring of integers in the totally real cubic subfield$K^{(49)}$ of the cyclotomic field $\mathbb{Q}(\zeta_7)$ has Pythagoras numberequal to $4$. This is the smallest possible value for a totally real numberfield of odd degree. Moreover, we determine which numbers are sums of integralsquares in this field, and use this knowledge to construct a diagonal universalquadratic form in five variables.

A cubic ring of integers with the smallest Pythagoras number