Math4302 Modern Algebra (Lecture 29)

Rings


R[x]=\{a_0+a_1x+\cdots+a_nx^n:a_0,a_1,\cdots,a_n\in R,n>1\}

Then (R[x],+,\cdot ) is a ring.

If R has a unity 1, then R[x] has a unity 1.

If R is commutative, then (R[x],+,\cdot ) is commutative.

Let F be a field, and F[x]. Fix \alpha\in F. \phi_\alpha:F[x]\to F defined by f(x)\mapsto f(\alpha) (the evaluation map).

Then \phi_\alpha is a ring homomorphism. \forall f,g\in F[x],

(f+g)(\alpha)=f(\alpha)+g(\alpha)
(fg)(\alpha)=f(\alpha)g(\alpha) (use commutativity of \cdot of F, f(\alpha)g(\alpha)=\sum_{k=0}^{n+m}c_k x^k, where c_k=\sum_{i=0}^k a_ib_{k-i})

Let \alpha\in F is zero (or root) of f\in F[x], if f(\alpha)=0.

Example

f(x)=x^3-x, F=\mathbb{Z}_3

f(0)=f(1)=0, f(2)=8-2=2-2=0

but note that f(x) is not zero polynomial f(x)=0, but all the evaluations are zero.

Division algorithm. Let F be a field, f(x),g(x)\in F[x] with g(x) non-zero. Then there are unique polynomials q(x),r(x)\in F[x] such that

f(x)=q(x)g(x)+r(x)

r(x) is the zero polynomial or \deg r(x)<\deg g(x).

Proof

Uniqueness: exercise

Existence:

Let S=\{f(x)-h(x)g(x):h(x)\in F[x]\}.