= Solution
The normal-error <log-likelihood>, for fixed <variance>, is a constant minus $\|Y-X\theta\|^2/(2\sigma^2)$. Differentiating the squared residual gives the normal equations $X^TX\widehat\theta=X^TY$. Since $X$ has full column rank, $X^TX$ is positive definite and the squared residual is strictly convex in $\theta$. Hence the unique minimizer, and thus the <maximum-likelihood estimator>, is
$$
\boxed{\widehat\theta=(X^TX)^{-1}X^TY.}
$$
Profiling over the unknown positive <variance> does not change the residual-minimizing value of $\theta$.
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