let $p$ be a prime number, $k$ a $cm$ field containing a primitive $p^(n)th$ root of unity and k the maximal real subfield of $k$. we show that a special case of a conjecture of solomon gives rise to a (conjectural) congruence mod $p^n$, relating certain $s$-units of $k$ to the principal semi-local units of $k$, via the use of local hilbert symbols. we will discuss the progress made in proving this conjecture and (briefly) some computational verifications.