$2$ is prime, but $2=(1+i)(1-i)$ is composite.

Question

If $2$ is a prime number, there is a corollary that says that every prime integer is irreducible on the Gaussian integers. But this is a contraction, for $2$ is composed of $(1+i)(1-i)$, and a number that can be composed cannot be irreducible.

How can this be?

I am totally confused. Does anyone propose to assist me?

Answer 1

Notice that $5 = \sqrt{5}\sqrt{5}$, so the prime $5$ can be factored in the ring $\mathbb{Z}[\sqrt{5}].$ "Prime" in this context means "can't be factored in the integers." It doesn't mean you can't add some more numbers to your system and create more factors.

Answer 2

That corollary probably says that every prime $p$ satisfying $p\equiv 3\pmod4$ stays prime among the Gauss integers.

Actually, every other primes (so $2$ and primes with $p\equiv1\pmod4$) do factor nontrivially in $\Bbb Z[i]$.