Theorem 11.9. Suppose f(x) is a polynomial of positive degree in Z[x] and p is a...

 

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Theorem 11.9. Suppose f(x) is a polynomial of positive degree in Z[x] and p is a prime number that does not divide the highest-degree coefficient of f(x). If the reduction [f](x) of f(x) modulo p is irreducible in Fp[x], then f(x) does not factor in Z[x] as a product of lower-degree polynomials. Exercise 11.16. Prove Theorem 11.9, using Theorem 11.8. Theorem 11.9 provides us with a new technique for proving that certain polynomials f(x) in Z[x] do not factor as a product of lower-degree polynomi- als: Find a prime number p that does not divide the highest-degree coefficient of f(x) (there are infinitely many from which to choose!) for which the reduc- tion [f](r) is irreducible in Fp[r]. The attraction of this technique resides in the fact that we can always determine, by the brute-force method of testing every possibility, whether a particular polynomial in F,[r] is irreducible. In particular, the irreducibility [f](x) in F,[x] may be easier to prove than the nonfactorizability of f(x) in Z[x] as a product of lower-degree polynomials. Let us consider two examples. Theorem 11.9. Suppose f(x) is a polynomial of positive degree in Z[x] and p is a prime number that does not divide the highest-degree coefficient of f(x). If the reduction [f](x) of f(x) modulo p is irreducible in Fp[x], then f(x) does not factor in Z[x] as a product of lower-degree polynomials. Exercise 11.16. Prove Theorem 11.9, using Theorem 11.8. Theorem 11.9 provides us with a new technique for proving that certain polynomials f(x) in Z[x] do not factor as a product of lower-degree polynomi- als: Find a prime number p that does not divide the highest-degree coefficient of f(x) (there are infinitely many from which to choose!) for which the reduc- tion [f](r) is irreducible in Fp[r]. The attraction of this technique resides in the fact that we can always determine, by the brute-force method of testing every possibility, whether a particular polynomial in F,[r] is irreducible. In particular, the irreducibility [f](x) in F,[x] may be easier to prove than the nonfactorizability of f(x) in Z[x] as a product of lower-degree polynomials. Let us consider two examples.

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To prove Theorem 119 we will use Theorem 118 which states that if fx is a polynomial of positive degree in Zx and p is a prime number that does not di... View the full answer