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**Laurent** If you apply $\displaystyle \tau_x$ to a decomposition $\displaystyle f(x)=C\prod_\rho (X-\rho)^{m_\rho}$, you get another decomposition $\displaystyle f(x)=\tau_x(f(x))=C\prod_\rho (X-\tau(\rho))^{m_\rho}$, hence by unicity of the decomposition into irreducible factors in $\displaystyle \overline{F}[x]$, $\displaystyle m_{\tau(\rho)}=m_\rho$ for every root $\displaystyle \rho$, and in particular the multiplicities of $\displaystyle \alpha$ and $\displaystyle \beta$ coincide. Does that look correct?