  This paper presents algebraic methods for the study of polynomial relative
invariants, when the group G formed by the symmetries and relative symmetries
is a compact Lie group. We deal with the case when the subgroup H of symmetries
is normal in G with index m, m greater or equal to 2. For this, we develop the
invariant theory of compact Lie groups acting on complex vector spaces. This is
the starting point for the study of relative invariants and the computation of
their generators. We first obtain the space of the invariants under the
subgroup $H$ of $\Gamma$ as a direct sum of $m$ submodules over the ring of
invariants under the whole group. Then, based on this decomposition, we
construct a Hilbert basis of the ring of G-invariants from a Hilbert basis of
the ring of H-invariants. In both results the knowledge of the relative
Reynolds operators defined on H-invariants is shown to be an essential tool to
obtain the invariants under the whole group. The theory is illustrated with
some examples.
