In perturbative quantum field theory several algebraic structures are used. Some for expanding Greens functions, others to establish renormalization. Basic features are the cyclicity of representations, the usage of creation and annihilation operators (field operators), and a state field correspondence. In mathematical terms this allows to generate the field algebra as a polynomial algebra (formal power series algebra) in the generators (one particle field operators). This implements a natural Hopf algebra structure on this ring and establishes certain submodules of `harmonic elements'. Symmetries of the quantum fields, being bosons, fermions or having internal structure, implement a subring structure which has a major impact on the algebraic structure under consideration. We claim in this talk, that (perturbative) quantum field theory is developed analogously to the theorem of Chevalley-Shepard-Todd-Bourbaki saying that a finite non-modular subgroup $G$ of $GL(V)$ has a freely generated polynomial invariant ring if the group $G$ is generated by pseudo reflections. This ring is the coordinate ring of the corresponding algebraic group scheme. We get several analogies e.g. the partition sum of pQFT is related to the Poincare series, quantization describes the fundamental pseudo reflections etc. Quantum field theory, if treated along this lines, can thus be rendered as non-commutative algebraic geometry. This is work in progress.