We study the minimum extension degree required for a Reed--Solomon code to admit exact $\mathbb{F}_q$-linear repair at the cut-set bound from every prescribed set of $d$ helpers. Let $s=d-k+1$, and let $Q_s(n)$ be the minimum product of $n$ pairwise coprime integers, each at least $s$. For every prime power $q$ and all $3\le k<d<n\le2d-2$, we prove that the minimum degree over $\mathbb{F}_q$ is exactly $sQ_s(n)$. When all surviving nodes are helpers, we also characterize the admissible extension degrees and bound the number of coordinates that can attain the cut-set bound. The required achievability follows from existing construction methods; we give a short existence proof using splitting subspaces. For a family of parameters with $n>2d-2$, we obtain strictly smaller extension degrees, showing that the range restriction cannot be omitted altogether. Finally, we prove that a nonlinear repair scheme attaining the cut-set bound induces an $\mathbb{F}_p$-linear repair scheme on the same code, where $p$ is the characteristic. Consequently, the extension-degree lower bounds over the prime field remain valid without a linearity assumption.

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