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Suppose $\\tilde{\\pi}_L$ is another uniformizer for $L$ such that $\\tilde{\\pi}_L\\equiv\\pi_L+r\\pi_L^{\\ell+1} \\pmod{\\pi_L^{\\ell+2}}$ for some $\\ell\\ge1$ and $r\\in O_K$. Let $\\tilde{f}(X)$ be the minimum polynomial for $\\tilde{\\pi}_L$ over $K$. In this paper we give congruences for the coefficients of $\\tilde{f}(X)$ in terms of $r$ and the coefficients of $f(X)$. 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Let $\\pi_L$ be a uniformizer for $L$ and let $f(X)$ be the minimum polynomial for $\\pi_L$ over $K$. Suppose $\\tilde{\\pi}_L$ is another uniformizer for $L$ such that $\\tilde{\\pi}_L\\equiv\\pi_L+r\\pi_L^{\\ell+1} \\pmod{\\pi_L^{\\ell+2}}$ for some $\\ell\\ge1$ and $r\\in O_K$. Let $\\tilde{f}(X)$ be the minimum polynomial for $\\tilde{\\pi}_L$ over $K$. In this paper we give congruences for the coefficients of $\\tilde{f}(X)$ in terms of $r$ and the coefficients of $f(X)$. 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