Rational Canonical and Jordan Forms

Definition 2.1. Suppose p1, . . . , pk are polynomials in F[x] which are not all 0. Set I = 〈p1, . . . , pk〉. Let d denote the monic generator of I. We call d the greatest common divisor of the pi and write d = gcd(p1, . . . , pk). If d = 1, we say that the polynomials p1, . . . , pk is relatively prime. Note that, by definition, there exists polynomials q1, . . . , qk ∈ F[x] such that d = p1q1 + · · · + pkqk. So, if p1, . . . , pk are relatively prime, we can find q1, . . . , qk such that ∑k i=1 piqi = 1. Lemma 2.2. Suppose p, q ∈ F[x] are not both 0. Set d = gcd(p, q). If e|p and e|q, then e|d. Proof. Write d = ap + bq with a, b ∈ F[x]. Then it is obvious.