Intereting Posts

Show $au_x+bu_y=f(x,y)$ gives $u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$ if $a\neq 0$.
Can I embed $\Bbb{C}(x)$ into $\Bbb{C}$?
Two different expansions of $\frac{z}{1-z}$
Grandi's Series; tends to $1/2$, but why is this considered a valid sum?
$C(n,p)$: even or odd?
How to find all irreducible polynomials in Z2 with degree 5?
How to transform a general higher degree five or higher equation to normal form?
When is $2^n \pm 1$ a perfect power
What is a type?
Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$
Is every irrational number normal in at least one base?
How to evaluate $\int_{0}^1 {\cos(tx)\over \sqrt{1+x^2}}dx$?
the converse of Schur lemma
Positive part of $y$ with $y\in L^2(0,T; H_0^1(\Omega))$ and $y'\in L^2(0,T; H^{-1}(\Omega))$
How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?

The following is a question from section $3.11$ of the book **An introduction to abstract algebra** by *Allenby*:

Explain intuitively why $\mathbb Z[\sqrt 2] \ncong \mathbb Z[\sqrt 3]$.back your intuition with proof.

Note:this example not only says that $\theta: a+b\sqrt 2 \mapsto a+b\sqrt 3$ is not isomorphism .It says no isomorphism can be found at all – no matter how clever choice of mapping you try to make..

- Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism
- Prove that $\Bbb{Z}/I$ is finite where I is an ideal of $\Bbb{Z}$
- Ring of order $p^2$ is commutative.
- If there is a unique left identity, then it is also a right identity
- The image of an ideal under a homomorphism may not be an ideal
- Is there any non-monoid ring which has no maximal ideal?

I can’t see what’s the intution behind this ..can anyone provide some hint on this…

- Least rational prime which is composite in $\mathbb{Z}$?
- Units in quotient ring of $\mathbb Z$
- Describe all ring homomorphisms
- Non-unital rings: a few examples
- Every maximal ideal is principal. Is $R$ principal?
- Modules with projective dimension $n$ have not vanishing $\mathrm{Ext}^n$
- Ideal in a ring of continuous functions
- How to prove $\mathbb{Z}=\{a+b\sqrt{2}i\mid a,b\in\mathbb{Z}\}$ is a principal ideal domain?
- If every commutator is idempotent, then the ring is commutative
- Isomorphism between quotient rings over finite fields

Hint: $2$ is a square in the first ring. Is it a square in the second?

- Embeddings are precisely proper injective immersions.
- Proving continuity of $f$
- bounded operator $T$ is not compact then there exists an orthonormal sequence $e_n$ and $d>0$ such that $\|T(e_n)\|>d$ for all $n\in\Bbb{N}$?
- About the integral $\int_{0}^{1}\frac{\log(x)\log^2(1+x)}{x}\,dx$
- If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?
- Irreducible polynomials have distinct roots?
- Find a number $\alpha > 1$ such that the following holds
- Topology ad Geometry of $\mathbb{C}^n/\mathbb{Z}_k$
- Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$
- Cardinality of Irrational Numbers
- Is the unit circle $S^1$ a retract of $\mathbb{R}^2$?
- Showing that $\mathbb{R}$ is connected
- If $f_n(x_n) \to f(x)$ whenever $x_n \to x$, show that $f$ is continuous
- Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
- Path-connected and locally connected space that is not locally path-connected