Abstract Algebra Theory and Applications - Computer Science ...

15.3 IRREDUCIBLE POLYNOMIALS 265p(x) and q(x). We have just shown that there exist polynomials u(x) andv(x) in F [x] such that d(x) = d ′ (x)[r(x)u(x) + s(x)v(x)]. Sincedeg d(x) = deg d ′ (x) + deg[r(x)u(x) + s(x)v(x)]and d(x) and d ′ (x) are both greatest common divisors, deg d(x) = deg d ′ (x).Since d(x) and d ′ (x) are both monic polynomials of the same degree, it mustbe the case that d(x) = d ′ (x).□Notice the similarity between the proof of Proposition 15.7 and the proofof Theorem 1.4.15.3 Irreducible PolynomialsA nonconstant polynomial f(x) ∈ F [x] is irreducible over a field F iff(x) cannot be expressed as a product of two polynomials g(x) and h(x)in F [x], where the degrees of g(x) and h(x) are both smaller than the degreeof f(x). Irreducible polynomials function as the “prime numbers” ofpolynomial rings.Example 4. The polynomial x 2 − 2 ∈ Q[x] is irreducible since it cannot befactored any further over the rational numbers. Similarly, x 2 +1 is irreducibleover the real numbers.Example 5. The polynomial p(x) = x 3 + x 2 + 2 is irreducible over Z 3 [x].Suppose that this polynomial was reducible over Z 3 [x]. By the divisionalgorithm there would have to be a factor of the form x − a, where a is someelement in Z 3 [x]. Hence, it would have to be true that p(a) = 0. However,p(0) = 2p(1) = 1p(2) = 2.Therefore, p(x) has no zeros in Z 3 and must be irreducible.Lemma 15.8 Let p(x) ∈ Q[x]. Thenp(x) = r s (a 0 + a 1 x + · · · + a n x n ),where r, s, a 0 , . . . , a n are integers, the a i ’s are relatively prime, and r and sare relatively prime.