# Why doesn't work Integer factorization for fields?

I try to unterstand, why the Integer factorization is only working for rings and not for fields. My first idea was, that you don't have a uniqueness quantification for prime "numbers" in fields. Is that correct? It would be nice if someone would explain it to me.

Greetings.

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There are no prime numbers in a field. –  André Nicolas Oct 26 '12 at 7:07

There are no prime numbers in a field. Every non-zero element of a field is a unit, where a unit is defined as a number $a$ which has a multiplicative inverse. A prime $p$ is in general defined as an object which is neither $0$ nor a unit, and such that whenever $p$ divides $ab$, $p$ divides $a$ or $p$ divides $b$. An object $p$ is irreducible if $p$ is neither $0$ nor a unit, and whenever $p=ab$, one of $a$ or $b$ is a unit. So by definition there are neither primes nor irreducibles in a field.