splitting fields of shifted generic polynomials

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I have come across the following expression in some research papers when they want to show that splitting fields are disjoint.



Let $K$ be an algebraically closed field of characteristic $p>0$, let $n$ be an odd integer number, let $f(X) = X^n + T_1 X^n-1 + cdots + T_n$, be the generic polynomial, that is, $T = (T_1, ldots, T_n)$ is a tuple of algebraically independent variables over $K$.



Let $Omega = $ $omega_1, ldots, omega_m$ be a finite subset of $K$, let $f_i(X) = f(X) - omega_i$, and let $F_i$ be the splitting field of $f_i$ over $K(T)$ ($i=1,ldots, m$).



Question: 1) Here why one has to calculate splitting fields of $f(x)-omega_i?$



2)If $F_i$ are linearly disjoint over $K(T)$, what do we conclude about the splitting field of $f(X)$




polynomials galois-theory symmetric-functions




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    I have come across the following expression in some research papers when they want to show that splitting fields are disjoint.



    Let $K$ be an algebraically closed field of characteristic $p>0$, let $n$ be an odd integer number, let $f(X) = X^n + T_1 X^n-1 + cdots + T_n$, be the generic polynomial, that is, $T = (T_1, ldots, T_n)$ is a tuple of algebraically independent variables over $K$.



    Let $Omega = $ $omega_1, ldots, omega_m$ be a finite subset of $K$, let $f_i(X) = f(X) - omega_i$, and let $F_i$ be the splitting field of $f_i$ over $K(T)$ ($i=1,ldots, m$).



    Question: 1) Here why one has to calculate splitting fields of $f(x)-omega_i?$



    2)If $F_i$ are linearly disjoint over $K(T)$, what do we conclude about the splitting field of $f(X)$




    polynomials galois-theory symmetric-functions




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I have come across the following expression in some research papers when they want to show that splitting fields are disjoint.



      Let $K$ be an algebraically closed field of characteristic $p>0$, let $n$ be an odd integer number, let $f(X) = X^n + T_1 X^n-1 + cdots + T_n$, be the generic polynomial, that is, $T = (T_1, ldots, T_n)$ is a tuple of algebraically independent variables over $K$.



      Let $Omega = $ $omega_1, ldots, omega_m$ be a finite subset of $K$, let $f_i(X) = f(X) - omega_i$, and let $F_i$ be the splitting field of $f_i$ over $K(T)$ ($i=1,ldots, m$).



      Question: 1) Here why one has to calculate splitting fields of $f(x)-omega_i?$



      2)If $F_i$ are linearly disjoint over $K(T)$, what do we conclude about the splitting field of $f(X)$




      polynomials galois-theory symmetric-functions




      share|cite|improve this question











      I have come across the following expression in some research papers when they want to show that splitting fields are disjoint.



      Let $K$ be an algebraically closed field of characteristic $p>0$, let $n$ be an odd integer number, let $f(X) = X^n + T_1 X^n-1 + cdots + T_n$, be the generic polynomial, that is, $T = (T_1, ldots, T_n)$ is a tuple of algebraically independent variables over $K$.



      Let $Omega = $ $omega_1, ldots, omega_m$ be a finite subset of $K$, let $f_i(X) = f(X) - omega_i$, and let $F_i$ be the splitting field of $f_i$ over $K(T)$ ($i=1,ldots, m$).



      Question: 1) Here why one has to calculate splitting fields of $f(x)-omega_i?$



      2)If $F_i$ are linearly disjoint over $K(T)$, what do we conclude about the splitting field of $f(X)$





      polynomials galois-theory symmetric-functions





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 25 at 5:02









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