  We measure the relationship between stellar mass and stellar metallicity, the
stellar mass--metallicity relation (MZR), for 1336 star-forming galaxies at
$1.6\le z\le3.0$ (<z>=2.2) using rest-frame far-ultraviolet spectra from the
zCOSMOS-deep survey. High signal-to-noise composite spectra containing stellar
absorption features are fit with population synthesis model spectra of a range
of metallicity. We find stellar metallicities, which mostly reflect iron
abundances, scaling as
$(Z_{Fe,\ast}/Z_{Fe,\odot})=-(0.81\pm0.01)+(0.32+0.03)\log(M_\ast/10^{10}M_\odot)$
across the mass range of $10^9\lesssim M_\ast/M_\odot\lesssim10^{11}$, being
$\approx6\times$ lower than seen locally at the same masses. The instantaneous
oxygen-to-iron ratio ($\alpha$-enhancement) inferred using the gas-phase oxygen
MZRs, is on average found to be [O/Fe]$\approx0.47$, being higher than the
local [O/Fe]$\approx0$. The observed changes in [O/Fe] and [Fe/H] are
reproduced in simple flow-through gas-regulator models with steady
star-formation histories (SFHs) that follow the evolving main sequence. Our
models show that the [O/Fe] is determined almost entirely by the instantaneous
specific star formation rate alone while being independent of the SFHs, mass,
and the gas-regulation characteristics of the systems. We find that the
locations of $\sim10^{10}M_\odot$ galaxies at z~2 in the [O/Fe]--metallicity
planes are in remarkable agreement with the sequence of low-metallicity
thick-disk stars in our Galaxy. This manifests a beautiful concordance between
the results of Galactic archaeology and observations of high-redshift Milky Way
progenitors. However, there remains a question of how and when the old
metal-rich, low-$\alpha$/Fe stars seen in the bulge had formed by z~2 because
such a stellar population is not seen in our data and difficult to explain in
the context of our models.
