Let $\mathbb{Z}[i] = \{a+bi : a,b \in \mathbb{Z}\}$ and $\mathbb{Q}(i) = \{a+bi : a,b \in \mathbb{Q}\}$.
Find $\alpha \in \mathbb{Z}[i]$ such that $(3+5i,1+3i) = (\alpha)$
Since $\mathbb{Z}[i]$ is a Euclidean domain, every ideal is principal, so $\alpha$ exists and it is a greatest common divisor of $3+5i$ and $1+3i$. You could find it with the Euclidean algorithm, but also factorization is possible.
Suppose we know $\alpha$; then $3+5i=\alpha\beta$ for some $\beta$ and therefore $$ (3+5i)(3-5i)=34=\alpha\bar{\alpha}\beta\bar{\beta} $$ Since $34=2\cdot 17$, is it possible to know the prime factors of $3+5i$? If yes, how rigouruously could I do that?
The norm is multiplicative, so if you can factor $3+5i$ the factors must have norms $2$ and $17$. The only choices (up to units) with norm $2$ are $1+i, 2$ and $2$ is obviously not right, so try $1+i$. You get $3+5i=(1+i)(4+i)$