  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 number
equal to $4$. This is the smallest possible value for a totally real number
field of odd degree. Moreover, we determine which numbers are sums of integral
squares in this field, and use this knowledge to construct a diagonal universal
quadratic form in five variables.
