Intereting Posts

Proving the Shoelace Formula with Elementary Calculus?
Why is $A_5$ a simple group?
Unique least square solutions
Prove there's a simple path of length $k$ in a simple graph $G$ where all the vertices have degree of at least $k$
Functional Analysis – Banach-Steinhaus theorem
How do you evaluate this limit? $\lim\limits_{x\to-\infty}\frac{2x^5+x^2}{\sqrt{2x^{10}+x^2}}$
Splitting of prime ideals in algebraic extensions
Meaning of floor symbols in $f(x) = \dfrac{1}{2 \lfloor x \rfloor – x + 1}$
Analytic map with two fixed points on a simply connected domain is the identity
$\operatorname{Ext}$ and injectives, respectively projectives
Martingale and Submartingale problem
Limits of 2 variable functions
Equation that can easily be changed to output the digit in 1's, 10's ,100's etc?
Conservation of mass in hyperbolic PDE
Are there useful criterions whether a positive integer is the difference ot two positive cubes?

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..

- Must an ideal generated by an irreducible element be a maximal ideal?
- Can we do long division in $\mathbb Z$, where $n$ is square free?
- Gaussian Integers and Quotient Rings
- Question about all the homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}$
- $1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
- Showing that a ring is a field as well for one of the provided choices.

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

- show that if $K$ is a field then $K$ is principal
- What do prime ideals in $k$ look like?
- Nontrivial subring with unity different from the whole ring?
- Why is the localization at a prime ideal a local ring?
- Prove that $\mathbb Q2]$ is a field
- Showing a Ring of endomorphisms is isomorphic to a Ring
- Do the polynomial germs generate all the ring of germs?
- When each prime ideal is maximal
- For a ring $R$ with a single proper ideal $I$, show that $I$ is prime
- Counting diagonalizable matrices in $\mathcal{M}_{n}(\mathbb{Z}/p\mathbb{Z})$

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

- Must a Hermitian/Kähler Manifold have a complex structure?
- How to find out whether linear programming problem is infeasible using simplex algorithm
- Density of Gaussian Random variable conditioned on sum
- The set of convergence of a sequence of measurable functions is measurable
- Drawing subgroup diagram of Dihedral group $D4$
- Find $\sin\frac{\pi}{3}+\frac{1}{2}\sin\frac{2\pi}{3}+\frac{1}{3}\sin\frac{3\pi}{3}+\cdots$
- Vector path length of a hypotenuse
- If the absolute value of a function is continuous, is the function continuous?
- How to prove $P(A) \cup P(B) \subseteq P(A \cup B) $
- Exactness and Naturality
- Show that a vector that is orthogonal to every other vector is the zero vector
- Deduction Theorem Intuition
- Proving Irrationality
- clarify the term “arithmetics” when talking about Gödel's incompleteness theorems
- Units of a log of a physical quantity