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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."

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This is one way to figure out the "obvious" candidates:

You know that the prime ideal factors are ideals of norm $2$ and $17$. Such ideals must contain the numbers $2$ and $17$ respectively (since $17=0$ in the quotient).

So one factor has to contain $2$ and the other factor has to contain $17$. Of course both factors have to contain $1+\sqrt{-33}$.

So $(2,1+\sqrt{-33})$ and $(17,1+\sqrt{-33})$ are indeed good candidates to start with (You know that your factors have to contain those). And by just computing the quotient, you can ensure that those candidates are indeed maximal, hence you are done.

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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 $1 + \sqrt{-33}$.

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