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Math Help - Splitting field

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    Senior Member vincisonfire's Avatar
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    Splitting field

    Show that, for any nonconstant  p(x) \in F(x) , there is a field K containing F such that p(x) splits over K.
    I know that if q(x) is any nonconstant polynomial with coefficients from F , then there a field L containing F such that there is a root of q(x) in L.
    Now suppose p(x) is the product of irreducible polynomials  q_i(x) then we can take  L_i = F(x)/q_i(x) to be the field in which  q_i(x) has a root.
    My question is : would the direct sum of these fields be a field over which p(x) would split (supposing that in  L_i , q_i(x) splits )?
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    Quote Originally Posted by vincisonfire View Post
    Show that, for any nonconstant  p(x) \in F(x) , there is a field K containing F such that p(x) splits over K.
    I know that if q(x) is any nonconstant polynomial with coefficients from F , then there a field L containing F such that there is a root of q(x) in L.
    Now suppose p(x) is the product of irreducible polynomials  q_i(x) then we can take  L_i = F(x)/q_i(x) to be the field in which  q_i(x) has a root.
    My question is : would the direct sum of these fields be a field over which p(x) would split (supposing that in  L_i , q_i(x) splits )?
    The main result is that if f(x) is a non-constant polynomial in F there exists an extension field K such that there is \alpha \in K so that f(\alpha) = 0. Given a non-constant polynomial p(x) of degree n you can construct K_1 so that K_1 is an extension field that \alpha_1 \in K_1 with f(\alpha_1)=0. Think of p(x) as a polynomial in K_1 since it has \alpha_1 as a zero it means p(x) = (x-\alpha_1)p_1(x) where p_1(x)\in K_1[x]. But that means there is a field K_2 so that K_1\subseteq K_2 and there is \alpha_2\in K_2 with p_2(\alpha_2)=0. And so this means p(x) = (x-\alpha_1)(x-\alpha_2)p_2(x) if viewed as a polynomial in K_2. Continuing in this way we can construct, F\subseteq K_1\subseteq K_2\subseteq ... \subseteq K_n so that the entire polynomial (which has degree n) can be written into linear factors over K_n.
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