Factorising ideals in the ring of integers of a quadratic field In an undergraduate algebraic number theory course, I was given the question "If $K = \mathbb Q(\sqrt{-33})$ Factorise the ideal $(1+\sqrt{-33})\subset \mathcal O_K$ into a product of prime ideals." I know that the norm is  multiplicative and that the norm of $(1+\sqrt{-33})$ is $34=2\cdot 17$. Also all ideals $\mathfrak p$ of prime norm are prime ideals as $\left|\mathcal O_K/\mathfrak p \right|=p \implies \mathcal O_K/\mathfrak p \cong \mathbb Z/p\mathbb Z$ - a field. So $\mathfrak p$ is a maximal ideal and hence prime. So I'm looking for prime ideals of norm $2$ and $17$. I also want their product to contain $(1+\sqrt{-33})$. I was unsure of how to proceed from here, so I looked at the solutions. My lecturer writes "the obvious candidates are $(2,1+\sqrt{-33})$ and $(17, 1+\sqrt{-33})$." Why are these obvious? I managed to check by multiplying each of these ideals by their complex conjugate that they indeed have norm $2$ and $17$ respectively, but how can you tell immediately that these have the required norm. On the other hand, how do you know for sure that $1+\sqrt{-33}\in (2,1+\sqrt{-33})(17, 1+\sqrt{-33})$ before doing calculations? I know that I can say $1+\sqrt{-33}=17\cdot(1+\sqrt{-33})-8\cdot 2 \cdot(1+\sqrt{-33})\in  (2,1+\sqrt{-33})(17, 1+\sqrt{-33}),$ but to me that counts as calculation. He writes "$(1+\sqrt{-33})=(2,1+\sqrt{-33})(17, 1+\sqrt{-33})$ must work, but this can be verified using direct computation too."

For your second question: in a Dedekind domain $(I+J)(I \cap J) = IJ$, so
$$
(2, 1 + \sqrt{-33})(17, 1 + \sqrt{-33}) = (2, 1 + \sqrt{-33}) \cap (17, 1 + \sqrt{-33}) \ni 1 + \sqrt{-33}
$$
because $(2, 1 + \sqrt{-33})$ and $(17, 1 + \sqrt{-33})$ are different maximal ideals.
As for the other one, I have been taught to make very light use, if at all, of the word "obvious" in proofs. Anyway, first note that you need two ideals which lie over $2$ and $17$. Now, if you know that those ideals are prime, then they are indeed good candidates, because all the generators of the product are multiples of $1 + \sqrt{-33}$.
