Intereting Posts

Is every real valued function on an interval a sum of two functions with Intermediate Value Property?
Given vectors $u,v, i$, find vector $w$ such that $i$ is incenter of triangle $$
On monomial matrix (generalized permutation matrix )
Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?
intersection of two curves in a square
How to apply Thompson's A×B lemma to show this nice feature of characteristic p groups?
Properties of set $\mathrm {orb} (x)$
Two ambient isotopic curve segments, one has the length and the other does not
Matrix Multiplication – Product of and Matrix
Closed form for the integral $\int_0^\infty t^s/(1+t^2)$
What exactly is a trivial module?
Smallest nonhamiltonian 2-connected bicubic graph with chromatic index 3
If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$
Finding all possible $n\times n$ matrices with non-negative entries and given row and column sums.
Describe the image of the set $\{z=x+iy:x>0,y>0\}$ under the mapping $w=\frac{z-i}{z+i}$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need only look at the primes $2,3,5,7$. Moreover, because $x^2+57 \equiv x^2 \mod 3$, $x^2+57 \equiv x^2+2 \mod 5$, $x^2+57 \equiv x^2+1 \mod 7$, it can be seen that 2 and 3 split while 5 and 7 remain prime. We let $P\overline{P}=(2)$; it is a fact that $\langle P \rangle$ has order 2. Additionally, we let $Q\overline{Q}=(3)$.

Computing some norms, I have been unable to find anything useful; even doing a computer search did not yield any norms that had 2 and 3 as the only factors, so I could not produce any useful relations. I tried to use an intermediate prime number, but they ultimately did not tell me much.

However, I know for a fact that the answer is $\mathbb{Z}_2\times \mathbb{Z}_2$. The two facts that need to be shown for this to be true are that $\langle Q \rangle$ has order 2 and $\langle Q \rangle \neq \langle P \rangle$.

- Ideals of $\mathbb{Z}$ geometrically
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- Showing that if the initial ideal of I is radical, then I is radical.
- Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain?
- How to find the nilpotent elements of $\mathbb{Z}/(\prod p_i^{n_i})$?
- Kernel of the homomorphism $\mathbb C → \mathbb C$ deﬁned by $x→t,y→ t^{2},z→ t^{3}$.

Any hint (e.g. a useful norm) would be appreciated.

- When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?
- Generalization of Cayley-Hamilton
- Showing that $x^n -2$ is irreducible in $\mathbb{Q}$
- If there is a unique left identity, then it is also a right identity
- Pathologies in module theory
- Proving a basic property of polynomial rings
- How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p$?
- Characterization of ideals in rings of fractions
- Showing an ideal is a projective module via a split exact sequence
- Example of a commutative ring with identity with two ideals whose product is not equal to their intersection

You are right in saying that $2$ and $3$ split, but you can in fact go one step further and show that they both ramify.

Then $(2) = P^2$ for some prime ideal $P$, and $(3)=Q^2$ for some prime ideal $Q$. We also know that $\langle P \rangle$ and $\langle Q \rangle$ have order $2$ (otherwise $P$ and $Q$ are principal ideals dividing $(2)$ and $(3)$ respectively, but $2$ and $3$ do not admit any proper divisors in $R$).

All that is left therefore is to show that $\langle P \rangle \neq \langle Q \rangle$. Suppose that they are equal. Then $P = \lambda Q$ for some $\lambda \in \mathbb{C}^*$, which implies that $(2)=(3 \lambda^2)$ $-$ a contradiction.

Hence the class group is generated by two distinct elements of order $2$, and so it is $\mathbb{Z}_2 \times \mathbb{Z}_2$.

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- Weak convergence on a dense subset of test functions