Ring of Integers of Algebraic Number Field is UFD iff Class Number is 1

Theorem

Let $K$ be a field of algebraic numbers.

Let $\OO_K$ be the ring of integers of $K$.

Then $\OO_K$ is a unique factorization domain (UFD) if and only if the class number of $K$ is $1$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): class group
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): class group