Intereting Posts

Show that $\lim\limits_{n\to\infty}\sum\limits_{k=1}^n\frac{1}{n^2\log(1+\frac{k^2}{n^2})}=\frac{{\pi}^2}{6}$
Finding generators for products of ideals
Non-additive asymptotic upper density: $\mathsf{d}^\star(A\cup B) \neq \mathsf{d}^\star(A)+\mathsf{d}^\star(B)$
Omitting the hypotheses of finiteness of the measure in Egorov theorem
How to define addition through multiplication?
The Radon-Nikodym derivative of a measure such that $|\int f'\,d\mu|\le \|f\|_{L^2}$ for $f\in C^1$
Bijective local isometry to global isometry
Determine the matrix relative to a given basis
Hausdorff measure of rectifiable curve equal to its length
Infinite Factorization Power Series of $\sin(x)$
Is any type of geometry $not$ “infinitesimally Euclidean”?
Is the language of all strings over the alphabet “a,b,c” with the same number of substrings “ab” & “ba” regular?
Quotient Group G/G = {identity}?
Why is it impossible to define multiplication in Presburger arithmetic?
$L^2$ norm inequality

1/ How to count the number of elements of $\mathbb{Z}[i]/(1+2i)^n$?

2/ How to write $\mathbb{Z}[i]/(1+2i)^n$ as direct sum of cyclic groups (in view of the structure theorem of finite abelian groups)?

- Examples of loops which have two-sided inverses.
- Finding basis of $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ over $\mathbb{Q}$
- What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?
- Construct a finite field of order 27
- Show from the axioms: Addition in a quasifield is abelian
- Isomorphism from $B/IB$ onto $(B/I)$

- Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?
- Let K be a field, and $I=(XY,(X-Y)Z)⊆K$. Prove that $√I=(XY,XZ,YZ)$.
- Cyclic Automorphism group
- Show that number of solutions satisfying $x^5=e$ is a multiple of 4?
- What are the situations, in which any group of order n is abelian
- Finding the intermediate fields of $\Bbb{Q}(\zeta_7)$.
- Proving that $D_{12}\cong S_3 \times C_2$
- The number of Sylow subgroups on $G$ with $|G|=pqr$
- Show that image of $res$ lies in $H^n(H,A)^{G/H}$
- Intuition behind “ideal”

Write $(1+2i)^n=u+iv$, $u,v\in\mathbb Z\setminus\{0\}$. Since $\mathbb Z[i]\simeq\mathbb Z\times\mathbb Z$ (as $\mathbb Z$-modules), we have $$\mathbb Z[i]/(u+iv)\simeq \mathbb Z\times\mathbb Z/\langle(u,v),(-v,u)\rangle.$$ The Smith Normal Form of the matrix $\left(\begin{matrix}u&v\\-v&u \end{matrix}\right)$ is $\left(\begin{matrix}1&0\\0&u^2+v^2 \end{matrix}\right)$. (It is not hard to show that $\gcd(u,v)=1$.) Thus we get $$\mathbb Z[i]/(u+iv)\simeq \mathbb Z/(u^2+v^2)\mathbb Z.$$ This shows that $|\mathbb Z[i]/(u+iv)|=u^2+v^2$, and therefore $$u^2+v^2=N(u+iv)=N((1+2i)^n)=N(1+2i)^n=5^n.$$

Here’s one approach you could take. Note that $(1+2i)^n(1-2i)^n=5^n$ and $(1+2i)^n$ and $(1-2i)^n$ are coprime in $\mathbb{Z}[i]$, so by the Chinese remainder theorem, $$\mathbb{Z}[i]/5^n\cong \mathbb{Z}[i]/(1+2i)^n\times \mathbb{Z}[i]/(1-2i)^n$$ as rings. But $$\mathbb{Z}[i]/(1+2i)^n\cong \mathbb{Z}[i]/(1-2i)^n$$ as rings (though not as $\mathbb{Z}[i]$-modules) by sending $i$ to $-i$, and it is easy to see that the underlying abelian group of $\mathbb{Z}[i]/5^n$ is $\mathbb{Z}/5^n\times\mathbb{Z}/5^n$ (since an element of $\mathbb{Z}[i]/5^n$ is just a number $a+bi$ where both $a$ and $b$ are taken mod $5^n$). If you have any decomposition of $\mathbb{Z}/5^n\times\mathbb{Z}/5^n$ as a product of two isomorphic abelian groups, then both factors must be isomorphic to $\mathbb{Z}/5^n$ (this follows easily from the classification of finite abelian groups, for instance). Thus the additive group of $\mathbb{Z}[i]/(1+2i)^n$ must be isomorphic to $\mathbb{Z}/5^n$.

- Showing that if the initial ideal of I is radical, then I is radical.
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- Is there a systematic way to solve in $\bf Z$: $x_1^2+x_2^3+…+x_{n}^{n+1}=z^{n+2}$ for all $n$?
- Generating a random derangement
- On Galois groups and $\int_{-\infty}^{\infty} \frac{x^2}{x^{10} – x^9 + 5x^8 – 2x^7 + 16x^6 – 7x^5 + 20x^4 + x^3 + 12x^2 – 3x + 1}\,dx$
- Any even elliptic function can be written in terms of the Weierstrass $\wp$ function
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- How to evaluate $ \lim \limits_{n\to \infty} \sum \limits_ {k=1}^n \frac{k^n}{n^n}$?
- Find a plane that passes through a point and is perpendicular to 2 planes