  In this paper, we will prove a spectral theorem for self-adjoint compactoid
operators. Also, we will study the condition on which the coefficient field
must be imposed. In order to get the theorems, we will use the Fredholm theory
for compactoid operators. Moreover, the property of maximal complete field is
important for our study. These facts will allow us to find that the spectral
theorem depends only on the residue class field, and is independent of the
valuation group of the coefficient field. As a result, we can settle the
problem of the spectral theorem in the case where the residue class field is
formally real.
