1.3. The polynomial ring F[X].

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 I shall assume everyone knows the following (see Jacobson

Chapter II, or Math 593).

(a) Let I be a nonzero ideal in F[X]. If f(X) is a nonzero polynomial of least degree in I,

then I = (f(X)). When we choose f to be monic, i.e., to have leading coefficient one, it is

uniquely determined by I. There is a one-to-one correspondence between the nonzero ideals

of F[X] and the monic polynomials in F[X]. The prime ideals correspond to the irreducible

monic polynomials.

(b) Division algorithm: given f(X) and g(X) ∈ F[X] with g _= 0, we can find q(X) and

r(X) ∈ F[X] with deg(r) < deg(g) such that f = gq + r;mo reover, q(X) and r(X) are

uniquely determined. Thus the ring F[X] is a Euclidean domain.

(c) Euclid’s algorithm: Let f and g ∈ F[X] have gcd d(X);t he algorithm gives polynomials

a(X) and b(X) such that

a(X) ・ f(X) + b(X) ・ g(X) = d(X), deg(a) ≤ deg(g), deg(b) ≤ deg(f).

Recall how it goes. Using the division algorithm, we construct a sequence of quotients and

remainders:

f = q0g + r0

g = q1r0 + r1

r0 = q2r1 + r2

・ ・ ・

rn−2 = qnrn−1 + rn

rn−1 = qn+1rn.

Then rn = gcd(f, g), and

rn = rn−2 − qnrn−1 = rn−2 − qn(rn−3 − qn−1rn−2) = ・ ・ ・ = af + bg.

Maple knows Euclid’s algorithm—to learn its syntax, type “?gcdex;”.

(d) Since F[X] is an integral domain, we can form its field of fractions F(X). It consists

of quotients f(X)/g(X), f and g polynomials, g _= 0.

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