Characteristic of Ring of Integers Modulo Prime
Jump to navigation
Jump to search
Theorem
Let $\struct {\Z_p, +, \times}$ be the ring of integers modulo $p$, where $p$ is a prime number.
The characteristic of $\struct {\Z_p, +, \times}$ is $p$.
Proof
From Ring of Integers Modulo Prime is Field we have that $\struct {\Z_p, +, \times}$ is a field.
So Characteristic of Finite Ring with No Zero Divisors applies, and so the characteristic of $\struct {\Z_p, +, \times}$ is prime.
The result follows.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Example $23$