= Solution
At a critical point, a <local field operator> $\mathcal O$ has <scaling dimension> $\Delta_{\mathcal O}$ if $\mathcal O(x)\mapsto b^{-\Delta_{\mathcal O}}\mathcal O(x/b)$ under coarse-graining by $b$. The coupling $u$ in $\int d^dx\,u\mathcal O$ has renormalization-group <eigenvalue>
$$
y_u=d-\Delta_{\mathcal O}.
$$
A perturbation is a <relevant operator> when $y_u>0$, an <irrelevant operator> when $y_u<0$, and a <marginal operator> when $y_u=0$; nonlinear terms in its <renormalization-group beta function> decide the fate of a marginal coupling. Irrelevant microscopic interactions decay under coarse-graining, so many distinct systems approach the same fixed point and share one <universality class>.
Near criticality the singular <free energy> density is of order one per correlation volume:
$$
f_{\rm s}\sim\xi^{-d}\sim t^{\nu d}.
$$
The <heat capacity> contains two derivatives with respect to <temperature>, so
$$
c_{\rm s}\sim\frac{d^2f_{\rm s}}{dt^2}
\sim t^{\nu d-2}=t^{-\alpha}.
$$
Consequently the <hyperscaling relation> is
$$
\boxed{\alpha=2-d\nu.}
$$
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