# Math4302 Modern Algebra (Lecture 29)

## Rings

### Polynomial 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.

#### Definition of evaluation map

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 $c_k=\sum_{i=0}^k a_ib_{k-i}$)

#### Definition of roots

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

<details>
<summary>Example</summary>

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

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

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

</details>

#### Factorization of polynomials

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 $\deg r(x)<\deg g(x)$.

<details>
<summary>Proof</summary>

Uniqueness: exercise

---

Existence:

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

</details>