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Consider the polynomial f(x) = x p − x + 1 ∈ Zp[x]. (a) Let a be a root of f in some extension. Prove that a /∈ Zp and a + b is a root of f for all b ∈ Zp. (b) Prove that f is irreducible over Zp. [Hint: Assume it is reducible. If one of the factors has degree m, look at the coefficient of x m−1 and get a contradiction.]
Hello, I need help with all parts of Problem 13 (a and b). Please show all the steps and the solutions of the problem. Thank you very much. 13. (a) Let f(x , K a field. Form the NEW polynomial g(x) f(x1). Prove: If g(x) is irreducible in Kx] then f(x) is irreducible in K (b) Factor -1 EQL] into irreducible polynomials in Q[r. (Hint: First factor out a linear term arising from a root. Then use (a) to investigate...