Lemma 15.110.1. Let $\varphi : A \to B$ be a surjection of rings. Let $G$ be a finite group of order $n$ acting on $\varphi : A \to B$. If $b \in B^ G$, then there exists a monic polynomial $P \in A^ G[T]$ which maps to $(T - b)^ n$ in $B^ G[T]$.

**Proof.**
Choose $a \in A$ lifting $b$ and set $P = \prod _{\sigma \in G} (T - \sigma (a))$.
$\square$

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