Aluminium oxide (alumina; Al_2_O_3_) is a promising candidate as a primary dust condensate in the atmospheres of oxygen-rich evolved stars. Therefore, alumina 'seed' particles might trigger the onset of stellar dust formation and of stellar mass loss in the wind. However, the formation of alumina dust grains is not well understood. Aims. We aim to shed light on the initial steps of cosmic dust formation (i.e. nucleation) in oxygen-rich environments via a quantum- chemical bottom-up approach. Starting with an elemental gas-phase composition, we construct a detailed chemical-kinetic network that describes the formation and destruction of aluminium-bearing molecules and dust- forming (Al_2_O_3_)_n_ clusters up to the size of dimers (n=2) coagulating to tetramers (n=4). Intermediary species include the prevalent gas- phase molecules AlO and AlOH as well as Al_x_O_y_ clusters with x=1-5, y=1-6. The resulting extensive network is applied to two model stars, which represent a semi-regular variable and a Mira type, and to different circumstellar gas trajectories, including a non-pulsating outflow and a pulsating model. The growth of larger-sized (Al_2_O_3_)_n_ clusters with n=4-10 is described by the temperature-dependent Gibbs free energies of the most favourable structures (i.e. the global minima clusters) as derived from global optimisation techniques and calculated via density functional theory. We provide energies, bond characteristics, electrostatic properties, and vibrational spectra of the clusters as a function of size, n, and compare these to corundum, which corresponds to the crystalline bulk limit (n to infinity). The circumstellar aluminium gas-phase chemistry in oxygen- rich giants is primarily controlled by AlOH and AlO, which are tightly coupled by the reactions AlO+H_2_, AlO+H_2_O, and their reverse. Models of semi-regular variables show comparatively higher AlO abundances, as well as a later onset and a lower efficiency of alumina cluster formation when compared to Mira-like models. The Mira-like models exhibit an efficient cluster production that accounts for more than 90% of the available aluminium content, which is in agreement with the most recent ALMA observations. Chemical equilibrium calculations fail to predict both the alumina cluster formation and the abundance trends of AlO and AlOH in the asymptotic giant branch dust formation zone. Furthermore, we report the discovery of hitherto unreported global minimum candidates and low-energy isomers for cluster sizes n=7, 9, and 10. A homogeneous nucleation scenario, where Al2O3 monomers are successively added, is energetically viable. However, the formation of the Al2O3 monomer itself represents an energetic bottleneck. Therefore, we provide a bottom-up interpolation of the cluster characteristics towards the bulk limit by excluding the monomer, approximately following an n^(-1/3)^ dependence.