Arithmetic properties of $F[x]/(m(x))$

Let $F$ be a field. Is $F[x]/(m(x))$ a ring? Is there an analog of Euler's theorem 1.8.4? The answer to both is yes.

Proposition 2.6.1   Let $F$ be a field. For any polynomials $a(x),b(x),c(x)\in F[x]$ ,

1. $\overline{a(x)}+\overline{b(x)}= \overline{b(x)}+\overline{a(x)}$ , (``addition is commutative'')
2. $\overline{a(x)}\overline{b(x)}= \overline{b(x)}\overline{a(x)}$ , (``multiplication is commutative'')
3. $(\overline{a(x)}+\overline{b(x)})+\overline{c(x)} =\overline{a(x)}+(\overline{b(x)}+\overline{c(x)})$ , (``addition is associative'')
4. $(\overline{a(x)}\overline{b(x)})\overline{c(x)} =\overline{a(x)}(\overline{b(x)}\overline{c(x)})$ , (``multiplication is associative'')
5. $(\overline{a(x)}+\overline{b(x)})\overline{c(x)}= \overline{a(x)}\overline{c(x)}+ \overline{b(x)}\overline{c(x)}$ , (``distributive'')
6. $\overline{a(x)}\overline{1}=\overline{a(x)}$ , (`` $\overline{1}$ is a multiplicative identity'')
7. $\overline{a(x)}+\overline{-a(x)}=\overline{0}$ ,
8. $\overline{a(x)}\overline{0}=\overline{0}$ ,
9. $\overline{a(x)}+\overline{0}=\overline{a(x)}$ (`` $\overline{0}$ is an additive identity''),
10. if $\overline{a(x)}\overline{c(x)}= \overline{b(x)}\overline{c(x)}$ and $gcd(m(x),c(x))=1$ then $\overline{a(x)} =\overline{b(x)}$ (``cancellation law'').

Each of these properties of the congruence classes corresponds to a property of polynomials already proven in a previous section.

Example 2.6.2   Let $R=\mathbb{R}[x]/(x^2+1)$ . We have $1\equiv 1 ({\rm mod} x^2+1)$ , $x\equiv x ({\rm mod} x^2+1)$ , $x^2\equiv -1 ({\rm mod} x^2+1)$ , ... $x^n\equiv (-1)^n ({\rm mod} x^2+1)$ , $x^n\equiv x(-1)^n ({\rm mod} x^2+1)$ ). Therefore, any element $\overline{p(x)}\in R$ may be represented by a polynomial of degree $\leq 1$ . As a set, we have

$$R=\{a+xb \vert a,b\in \mathbb{R}\}.$$

Addition on $R$ corresponds to the usual addition of polynomials on $\{a+xb \vert a,b\in \mathbb{R}\}$ . Multiplication on $R$ corresponds to

$$(a+xb)(c+xd)=ac+bd+(ad+bc)x,$$

since $x^2=-1$ . In fact, with these definitions of addition and multiplication, $R$ is a field. Moreover, the map $\phi:R\rightarrow \mathbb{C}$ , defined by $\phi(a+xb)= a+ib$ , is an isomorphism of fields. In other words, under $\phi$ , addition and multiplication of elements of $R$ corresponds to addition and multiplication of elements of $\mathbb{C}$ in the sense that

$$\phi(a)\phi(b)=\phi(ab), \ \phi(a)+\phi(b)=\phi(a+b),$$

for all $a,b\in \mathbb{R}$ .