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If $f \in k[x]$ is the minimal polynomial of $\alpha \in K$, show how to write down the minimal polynomial of $\alpha - \beta$ for $\beta \in k$.

I've been playing around with the minimal polynomial of square roots in $\mathbb{Q}[x]$, for example comparing the minimal polynomials of $\sqrt{2}$ and $\sqrt{2}-1$... But I can't seem to find any generality.

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Hint. $f(X+\beta) \in k[x]$ has $\alpha-\beta$ as a root. Is it also irreducible/minimal?

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