Irreducible Polynomial

2020-2학기, 대학에서 ‘현대대수1’ 수업을 듣고 공부한 바를 정리한 글입니다. 지적은 언제나 환영입니다 :)

---

Definition. Irreducible Polynomial

Let $F$ be a field, and $f(x) \in F[x]$.

if $f(x) \ne g(x) h(x)$ for any non-constant poly-. $g(x), h(x) \in F[x]$,

then $f(x)$ is irreducible over $F[x]$.

Example.

• $x^2 + 1$ is irreducible over $\mathbb{R}$
• $x^2 + 1$ is irreducible over $\mathbb{C}$

---

Theorem.

Let $f(x) \in F[x]$, and $\deg (f(x)) = 2 \; \textrm{or} \; 3$.

$f(x)$ is reducible over $F$ $\iff$ $f(x)$ has a solution over $F$.

Theorem.

Let $f(x) \in \mathbb{Z}[x]$ and $r, s < \deg (f(x))$.

$f(x) = g(x)h(x)$ over $\mathbb{Q}[x]$ with $\deg(g(x)) = r$, $\deg(h(x)) = s$

$\iff$

$f(x) = g_1(x)h_1(x)$ over $\mathbb{Z}[x]$ with $\deg(g_1(x)) = r$, $\deg(h_1(x)) = s$

proof.

Let $f(x) = g(x)h(x)$, $g(x), h(x) \in \mathbb{Q}[x]$.

(Case 1)

(Case 2)