The ring F[x]/p(x) is a field iff p is irreducible

Dependencies:

  1. A field is an integral domain
  2. F[x]/p(x): A ring
  3. Degree of product of polynomials
  4. Polynomial GCD theorem

Let $F$ be a field. Let $p(x) \in F[x]-F$. Then $F[x]/p(x)$ is a field iff $p$ is irreducible.

Proof

Suppose $p$ is reducible. $\Rightarrow p = ab$, where $a, b \in F[x]-F$. Then $a \circ b = (ab)\%p = p\%p = 0$. This means $F[x]/p(x)$ has zero-divisors. Therefore, $F[x]/p(x)$ is not a field.

The unity of $F$ is also a unity of $F[x]/p(x)$.

Suppose $p$ is irreducible.

Let $a(x) \in F[x]/p(x)-\{0\} \Rightarrow 0 \le \deg(a) < \deg(p)$. Let $g = \gcd(a, p)$.

$g \mid a \Rightarrow \exists b \in F[x], a = gb$.

\begin{align} & a \neq 0 \\ &\Rightarrow (g \neq 0 \wedge b \neq 0) \\ &\Rightarrow (0 \le \deg(g) \wedge 0 \le \deg(b)) \\ &\Rightarrow 0 \le \deg(g) \le \deg(a) < \deg(p) \tag{$\deg(a) = \deg(g) + \deg(b)$} \end{align}

$g \mid p \Rightarrow \exists h \in F[x], p = gh$.

\[ p \neq 0 \Rightarrow (g \neq 0 \wedge h \neq 0) \]

\begin{align} & p \textrm{ is irreducible} \\ &\Rightarrow g \in F \vee h \in F \\ &\Rightarrow g \in F-\{0\} \vee h \in F-\{0\} \tag{$g \neq 0 \wedge h \neq 0$} \\ &\Rightarrow \deg(g) = 0 \vee \deg(h) = 0 \\ &\Rightarrow \deg(g) = 0 \vee \deg(g) = \deg(p) \tag{$\deg(p) = \deg(g) + \deg(h)$} \\ &\Rightarrow \deg(g) = 0 \tag{$\deg(g) \le \deg(a) < \deg(p)$} \\ &\Rightarrow g = 1 \tag{$g$ is monic} \end{align}

By the GCD theorem, $1 = \gcd(a, p) = sa + tp$ where $s, t \in F[x]$.

Let $s = kp + s\%p$. Then

\[ 1 = sa + tp = (s\%p + kp)a + tp = (s\%p)a + (ka+t)p \]

Therefore, we can assume without loss of generality that $s \in F[x]/p(x)$.

\[ (s \circ a) = (sa)\%p = (1 - tp)\%p = 1 \] Therefore, $a^{-1} = s$. Therefore, every element in $F[x]/p(x)-\{0\}$ has a multiplicative inverse. therefore, $F[x]/p(x)$ is a field.

Dependency for: None

Info:

Transitive dependencies:

  1. /polynomials/commutative
  2. Group
  3. Ring
  4. Polynomial
  5. GCD of polynomials
  6. Degree of product of polynomials
  7. Zero divisors of a polynomial
  8. Polynomial divisibility
  9. Degree of sum of polynomials
  10. Integral Domain
  11. 0x = 0 = x0
  12. Field
  13. A field is an integral domain
  14. Polynomial division theorem
  15. Polynomial GCD theorem
  16. Identity of a group is unique
  17. Subgroup
  18. Inverse of a group element is unique
  19. Conditions for a subset to be a subgroup
  20. Condition for a subset to be a subgroup
  21. F[x]/p(x): A ring