  We derive B\'ezout identities for the minimal polynomials of a finite
sequence and use them to prove a theorem of Wang and Massey on binary sequences
with a perfect linear complexity profile. We give a new proof of Rueppel's
conjecture and simplify Dai's original proof. We obtain short proofs of results
of Niederreiter relating the linear complexity of a sequence s and K(s), which
was defined using continued fractions. We give an upper bound for the sum of
the linear complexities of any sequence. This bound is tight for sequences with
a perfect linear complexity profile and we apply it to characterise these
sequences in two new ways.
