Intereting Posts

Another version of the Poincaré Recurrence Theorem (Proof)
Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$
Finding sum $\sum_{i=1}^n \frac1{4i^2-1}$
connected sum of torus with projective plane
Simple algebra question – proving $a^2+b^2 \geqslant 2ab$
Probability question about married couples
$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$
Manifolds with geodesics which minimize length globally
Tricky positive diophantine equation
Ideas about Proofs
A generalization of the connected sum of links
Maximising sum of sine/cosine functions
Show that $\int_0^\infty\frac{1}{x ((\ln x)^2+1)^p}dx$ converges for any $p\geq 1$ and find its value.
When is $c^4-72b^2c^2+320b^3c-432b^4$ a positive square?
Calculate $\mathbb{E}(W_t^k)$ for a Brownian motion $(W_t)_{t \geq0}$ using Itô's Lemma

I am working with the ideals $\mathfrak{p}=\left<2,1+\sqrt{-5}\right>, \mathfrak{q}=\left<3,1+\sqrt{-5}\right>, \mathfrak{t}=\left<3,1-\sqrt{-5}\right>$ and I am trying to prove that they are prime in $\mathbb{Z}[\sqrt{-5}]$.

I understand a good method to do this involves taking norms of each element that generates the ideals. This gives, for instance, the norms of the generators of $\mathfrak{p}=4,6$. These are both divisible by $2$. I have also calculated $\mathfrak{p}^2=\left<2\right>$. I get the feeling that since both norms are divisible by the generator of the ideal squared this proves it is prime, but I am not quite sure why.

In a similar vein, the norms of the generators of $\mathfrak{q,t}=9,6$ which are both divisible by $3$. I know $\mathfrak{qt}=\left<3\right>$, but I am not sure if this is relevant, since it is not either ideal squared. I think the norm of the ideal itself being prime implies the ideal is prime, but I am not sure how to find the norm of the ideal from the norm of its generating elements.

- Why is it called a 'ring', why is it called a 'field'?
- What specific algebraic properties are broken at each Cayley-Dickson stage beyond octonions?
- Artinian ring and faithful module of finite length
- Finding all homomorphisms between two groups - couple of questions
- linear algebra over a division ring vs. over a field
- If $G$ is a finite group of order $n$, then $n$ is the minimal such that $g^n=1$ for all $g \in G$?

This question is similar to: Prove that ideals are prime, but I don’t quite understand the reasoning behind the chosen answer.

I am not sure how the answerer deduces that the example there is prime either. Following their reasoning as far as I can, the fact that $2$ divides both norms means that $\left<2\right> \subset \mathfrak{p} \subset \mathbb{Z}[\sqrt{-5}]$ but I’m not sure how, and then I don’t know how this proves it is prime.

- Are $C(\mathbb{R})$ and $D(\mathbb{R})$ isomorphic or not?
- Motivation behind the definition of ideal class group
- A proof about polynomial division
- Is there a ring homomorphism $M_2(\mathbb Z)\to \mathbb Z$?
- Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
- Units and Nilpotents
- P p-sylow with $ P ⊂ Z(G) $
- Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.
- Do groups of order $p^3$ have subgroups of order $p^2$?
- Does multiplying polynomials ever decrease the number of terms?

There are several ways to deduce that these are ideals are prime. The easiest might be to just compute the quotient:

$$\mathbb Z[\sqrt{-5}]/(2,1+\sqrt{-5}) \cong \mathbb Z[X]/(X^2+5,2,1+X) = \mathbb Z[X]/(2,1+X) \cong \mathbb Z/2\mathbb Z$$

But you can also use some theory (And this somehow fits to your norm approach). Whenever we have an quadratic integer ring and an integer prime number $p \in \mathbb Z$, then the ideal $(p) \subset \mathcal O_K$ behaves in three ways:

- $(p)$ is prime.
- $(p) = \mathfrak p^2$ for some prime ideal.
- $(p) = \mathfrak p_1 \mathfrak p_2$ for two different prime ideals.

Together with the fact, that there exists a unique prime ideal factorization, we get the following corollary: Whenever we have $(p)=IJ$ for some ideals $I,J$, then $I$ and $J$ are necessarily prime (If one of them would factor into primes, $(p)$ would factor into at least $3$ primes).

The norm of $\mathfrak{p}$ is

\begin{equation*}

\big\lvert \mathbb{Z}[\sqrt{-5}]\big/(2,1+\sqrt{-5})\big\rvert = 2,

\end{equation*}

as was observed in another answer. So if $\mathfrak{p}$ is a product of two ideals, their norms must have product $2$. Thus one of the ideals has norm $1$ so is the whole ring. Thus $\mathfrak{p}$ is prime.

In that answer, the answerer speaks about norm of ideals, not norm of numbers. You can think geometrically, if $\langle 2, 1 + \sqrt{-5} \rangle \subset A$ then its lattice is a sub lattice of $A$, so the area of the fundamental area (the area of the smallest parallelogram in the lattice ) of $A$ divide the area of the fundamental area of $\langle 2, 1 + \sqrt{-5}\rangle$ but this area is two so the fundamental area of $A$ is one and $A = \mathbb{Z}[\sqrt{-5}]$. So $\langle 2, 1 + \sqrt{-5}\rangle$ is a maximal ideal, and so it is a prime ideal.

- How do I prove that a function with a finite number of discontinuities is Riemann integrable over some interval?
- What is stopping every Mordell equation from having a elementary proof?
- How to find limit of the sequence $\sum\limits_{k=1}^n\frac{1}{\sqrt {n^2 +kn}}$?
- Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
- Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture
- Trouble in understanding a proof of a theorem related to UFD.
- Why do the rationals, integers and naturals all have the same cardinality?
- Finding sum $\sum_{i=1}^n \frac1{4i^2-1}$
- Describing a Wave
- Finding XOR of all subsets
- Symbols for Quantifiers Other Than $\forall$ and $\exists$
- What is the $x$ in $\log_b x$ called?
- How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 6?
- Are there real world applications of finite group theory?
- How can I visualize a four-dimensional point inside a Schlegel diagram of a tesseract?