  The linear complexity of a periodic sequence over $GF(p^m)$ plays an
important role in cryptography and communication [12]. In this correspondence,
we prove a result which reduces the computation of the linear complexity and
minimal connection polynomial of a period $un$ sequence over $GF(p^m)$ to the
computation of the linear complexities and minimal connection polynomials of
$u$ period $n$ sequences. The conditions $u|p^m-1$ and
  $\gcd(n,p^m-1)=1$ are required for the result to hold. Some applications of
this reduction in fast algorithms to determine the linear complexities and
minimal connection polynomials of sequences over $GF(p^m)$ are presented.
