  When the quantum parameter $q^{\frac{1}{2}}$ is a root of unity of odd order
and the punctured bordered surface has nonempty boundary, we prove the fraction
ring of the stated skein algebra (that is the localization over all nonzero
elements) is a symmetric Frobenius algebra over both the field of fractions of
the image of the Frobenius map and the field of fractions of the center of the
stated skein algebra. We also calculate Traces of the fraction ring of the
stated skein algebra over these two fields.
