The Characteristic of a Ring

The Characteristic of a Ring

Definition: Let $R$ be a ring. The Characteristic of $R$ denoted $\mathrm{char} (R)$ or $\mathrm{ch}(R)$ is the smallest nonnegative $p$ such that $p \cdot 1 = 0$. If no such $p$ exists then we define the $\mathrm{char} (R) = 0$.

For example, consider the fields $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$. It is easy to see that the characteristic of each of these fields is $0$, for clearly $p \cdot 1 = p = 0$ if equal to $0$ if and only if $p = 0$.

For another example, consider the Galois field $GF(p^n)$. It is easy to verify that the characteristic of this field is $p$.

The following theorem tells us that the characteristic of a field is always a prime number of $0$.

Theorem 1: Let $F$ be a field. Then $\mathrm{char}(F) = p$ for some prime $p$ or $\mathrm{char}(F) = 0$.
  • Proof: Suppose that $\mathrm{char}(F) = n$ for some $n \in \mathbb{N}$. Then:
\begin{align} \quad n \cdot 1 = 0 \end{align}
  • Suppose that $n$ is a composite number, say $n = rs$. Then:
\begin{align} \quad rs \cdot 1 &= 0 \\ \quad (r \cdot 1) \cdot (s \cdot 1) &= 0 \end{align}
  • Since $F$ is a field $F$ has no zero divisors. So $r \cdot 1 = 0$ or $s \cdot 1 = 0$. But this contradicts $n$ being the least such nonnegative integer with this property. So $n = p$ for some prime $p$. $\blacksquare$
Theorem 2: Let $F$ be a field with $\mathrm{char}(F) = p$ for some prime $p$. Then $(a \pm b)^{p^n} = a^{p^n} \pm b^{p^n}$ for all $a, b \in F$ and for all $n \in \mathbb{N}$.
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