Exercise 6. Let F be a field, show that f(x) F[x] is irreducible if and...

 

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Exercise 6. Let F be a field, show that f(x) € F[x] is irreducible if and only if f(x + r) is irreducible for all r ER. Note that by the binomial theorem, (x + r) is the following polynomial. k (x + r) ² = [ (i) Σ(1) k and so if f(x) = Cnxk + • + C₁x + Co, we have pk-ipi f(x+r) = C₂(x + r)² + ··· + c₁(x + r) + co as a polynomial in R[x]. But seriously, you do not need the above expression, it only helps you understand the animal you are facing. The real proof is short. Exercise 6. Let F be a field, show that f(x) € F[x] is irreducible if and only if f(x + r) is irreducible for all r ER. Note that by the binomial theorem, (x + r) is the following polynomial. k (x + r) ² = [ (i) Σ(1) k and so if f(x) = Cnxk + • + C₁x + Co, we have pk-ipi f(x+r) = C₂(x + r)² + ··· + c₁(x + r) + co as a polynomial in R[x]. But seriously, you do not need the above expression, it only helps you understand the animal you are facing. The real proof is short.

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Solution suppose that fx is a polynomial in the Fx which is reducible th... View the full answer