= Solution
The <Gaussian likelihood> has <log-likelihood>, apart from no omitted parameter-dependent terms,
$$
\ell(\beta,\sigma^2)
=-\frac n2\log(2\pi)-\frac n2\log\sigma^2
-\frac{1}{2\sigma^2}(Y-X\beta)^T(Y-X\beta).
$$
For fixed $\sigma^2$, maximizing it is <ordinary least squares>. Differentiating with respect to $\beta$ gives the <normal equation> $X^TX\widehat\beta=X^TY$. <Full column rank> makes the quadratic strictly convex, so
$$
\boxed{\widehat\beta=(X^TX)^{-1}X^TY.}
$$
At this value, differentiating with respect to $v=\sigma^2$ gives $-n/(2v)+\mathrm{RSS}/(2v^2)=0$, so
$$
\boxed{\widehat\sigma^2_{\mathrm{ML}}=\frac{\|Y-X\widehat\beta\|^2}{n}.}
$$
For positive RSS this is the global likelihood maximum in $v>0$. RSS is positive almost surely under the model since $p<n$; an exactly zero observed RSS instead gives a boundary supremum as $v\downarrow0$.
As a <linear image of a multivariate normal vector>, the coefficient estimate satisfies
$$
\boxed{\widehat\beta\sim N_p\!\left(\beta,\sigma^2(X^TX)^{-1}\right).}
$$
Indeed, $\widehat\beta-\beta=(X^TX)^{-1}X^T\varepsilon$, and direct matrix multiplication gives its <covariance matrix>.
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