Since A = (-5)^(1/5) is in R f(x) is not irreducible, f(x) = (x-A)(X^4+Ax^3+A^2x^2+A^3x+A^4).
If B is fifth root of 1, i.e, B^5 =1 then f(x) = (x-A)(x-AB)(x-AB^2)(x-AB^3)(x-AB^4) splits completely in C (the fundamental theorem of algebra states that C is an algebraic closure so all polynomials split completely in C http://en.wikipedia.org/wiki/Fundamental_theorem_o... )
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By Eisenstein's criterion it's irreducible over Q (http://en.wikipedia.org/wiki/Eisenstein%27s_criter... ).
Since A = (-5)^(1/5) is in R f(x) is not irreducible, f(x) = (x-A)(X^4+Ax^3+A^2x^2+A^3x+A^4).
If B is fifth root of 1, i.e, B^5 =1 then f(x) = (x-A)(x-AB)(x-AB^2)(x-AB^3)(x-AB^4) splits completely in C (the fundamental theorem of algebra states that C is an algebraic closure so all polynomials split completely in C http://en.wikipedia.org/wiki/Fundamental_theorem_o... )