  In this paper, we mainly give a general explicit form of Cassels' $p$-adic
embedding theorem for number fields. We also give its refined form in the case
of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over
$Z$, we give a general unconditional upper bound for the smallest prime number
$p$ such that $f$ has a simple root modulo $p$.
