= Solution
{c}
For a <Dirac field>, the <path integral> is a <Berezin integral> over independent <Grassmann fields> $\psi$ and $\bar\psi$:
$$
\boxed{Z=\mathcal N\int\mathcal D\bar\psi\,\mathcal D\psi\,
\exp\left[i\int d^4x\,\bar\psi(i\not\partial-m)\psi\right].}
$$
The vacuum <boundary conditions> and the <Feynman i-epsilon prescription> are understood; $\mathcal N$ fixes the vacuum normalization. This is the vacuum <generating functional> without sources, rather than a finite-temperature <partition function>.
The <Dirac adjoint> transforms as $\bar\psi\mapsto\bar\psi e^{-i\alpha}$. For constant $\alpha$, $\partial_\mu(e^{i\alpha}\psi)=e^{i\alpha}\partial_\mu\psi$. Both the kinetic term and the <Dirac mass term> therefore retain their values. This proves the <global phase symmetry>, whose <group> is the <unitary group> $U(1)$.
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