  We compute the $O(\alpha_s \alpha^2)$ and $O(\alpha_s^2 \alpha)$
contributions to the production cross section of a $Z$ boson with one $b$ jet
at the Large Hadron Collider (LHC), and study their phenomenological relevance
for LHC physics. The accurate prediction of hadronic $Z+b$-jet production is
needed to control a background that greatly affects both the measurement of
Higgs-boson properties and searches of new physics at the LHC. At the same time
it could enable the first precise measurement of the $b$-quark parton
distribution function. In this context $b$-quark mass effects become relevant
and need to be studied with care, both at the level of the hard process and at
the level of the initial- and final-state parton evolution. It is the aim of
this paper to explore some of these issues in the framework of a massive 5
Flavor Scheme and to assess the need for both the inclusion of electroweak
corrections, in addition to QCD corrections, and $b$-quark mass effects in the
prediction of total and differential cross sections for hadronic $Z+b$-jet
production.
