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Abstract

In this thesis, the first high-precision measurement of the ground-state $g$-factor of a boronlike ion, $^{40}$Ar$^{13+}$, with a fractional uncertainty of \SI{1.4e-9}{}, is presented. The measurement has been performed on a single boronlike argon ion with the double Penning-trap setup of the newly developed ALPHATRAP experiment. Within this work, the trap tower of the experiment has been developed, assembled and tested prior to commissioning it together with the rest of the ALPHATRAP setup. The resulting measurement presented here corresponds to the most precise $g$-factor determination of a five-electron system to date. Not only does it allow testing the currently available theoretical predictions for the many-electron, QED and nuclear-recoil contributions, but also distinguishes between calculations that are in disagreement. The $g$-factor obtained here is in agreement with the most recent and most precise theoretical prediction, which has a relative uncertainty of \SI{9e-7}{}. This level of agreement constitutes one of the most accurate tests of many-electron QED contributions in strong fields. The sensitivity of this test will improve in the future with anticipated improvements on the theoretical $g$-factor, which includes higher-order QED contributions. Furthermore, this measurement paves the way towards the independent determination of the fine-structure constant with heavier highly charged ions in ALPHATRAP, where a specific difference of the boron- and hydrogenlike ions' $g$-factors will be used to cancel nuclear structure effects.