Irreducible Elements of Ring of Integers

Theorem

Let $\struct {\Z, +, \times}$ be the ring of integers.

The irreducible elements of $\struct {\Z, +, \times}$ are the prime numbers and their negatives.

Proof

We have that Integers form Integral Domain.

Therefore the concept of irreducible is defined.

Let $p$ be a prime number.

By definition, the only divisors of $p$ are $1, -1, p, -p$.

From Units of Ring of Integers, $1$ and $-1$ are (the only) units of $\Z$.

From Associates are Unit Multiples, $p$ and $-p$ are (the only) associates of each other.

Hence the result, from the definition of irreducible.

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 29$. Irreducible elements: Example $57$