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. [Hi...

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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. [Hi...

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.]

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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. [Hi...

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Add Answer to: 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. [Hi...

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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. [Hi...