Intereting Posts

Evaluate the series $\lim\limits_{n \to \infty} \sum\limits_{i=1}^n \frac{n+2}{2(n-1)!}$
Deriving Maclaurin series for $\frac{\arcsin x}{\sqrt{1-x^2}}$.
Involutions and Abelian Groups
Singularities at infinity
If $\sum a_n^2 n^2$ converges then $\sum |a_n|$ converges
What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…
How to solve $\sin x +\cos x = 1$?
If $x_1<x_2$ are arbitrary real numbers, and $x_n=\frac{1}{2}(x_{n-2}+x_{n-1})$ for $n>2$, show that $(x_n)$ is convergent.
Linear Dependence Lemma
A maximal ideal is always a prime ideal?
Expected value of infinite sum
Norm-Euclidean rings?
product distribution of two uniform distribution, what about 3 or more
Evaluating the limit of $\sqrt{2+\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}$ when $n\to\infty$
Definition of the principal symbol of a differential operator on a real vector bundle.

Can we find a field $K$ and an endomorphism $f$ of $K$, such that $f$ is not trivial and $f$ is not surjective? In other words, can we find an endomorphism of $K$ which is not an automorphism?

- Centralizer/Normalizer of abelian subgroup of a finite simple group
- Spectrum of $\mathbb{Z}$
- Reconciling Different Definitions of Solvable Group
- If $$ and $$ are relatively prime, then $G=HK$
- Zero-dimensional ideals and finite-dimensional algebras
- Isomorphism between quotient rings over finite fields
- proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
- Presentations of subgroups of groups given by presentations
- Finding the ring of integers of $\mathbb Q$ with $\alpha^5=2\alpha+2$.
- What are the three non-isomorphic 2-dimensional algebras over $\mathbb{R}$?

Consider the Frobenius endomorphism of fields of characteristic $p$ given by

$$ x \to x^{p}$$

This is not always an automorphism. For example, the image of rational function field $\mathbb{F}_{p}(t)$ under the Frobenius endomorphism does not contain $t$.

Yes, lots; for example, the endomorphisms $F(x) \to F(x)$ fixing $F$ are precisely given by extending $x \mapsto \frac{p(x)}{q(x)}$ where $p, q$ are two nonzero polynomials of total degree at least $1$. Assuming WLOG that $\gcd(p, q) = 1$, this map is an automorphism if and only if $p = ax + b, q = cx + d$ where $ad – bc \neq 0$ (exercise).

On the other hand, if $K$ is a finite extension of its prime subfield, then any endomorphism of $K$ is an automorphism (exercise). These are precisely the number fields and the finite fields.

- What's wrong with this “backwards” definition of limit?
- Multiple choice question: Let $f$ be an entire function such that $\lim_{|z|\rightarrow\infty}|f(z)|$ = $\infty$.
- Homeomorphism between Space and Product
- Question about basis and finite dimensional vector space
- What is an odd prime?
- Help With Plugging in Values Distance Point to Ellipse
- Evaluating 'combinatorial' sum
- Intuitive explantions for the concepts of divisor and genus
- Infinite set and countable subsets
- $\sum x_{k}=1$ then, what is the maximal value of $\sum x_{k}^{2}\sum kx_{k} $
- An integral$\frac{1}{2\pi}\int_0^{2\pi}\log|\exp(i\theta)-a|\text{d}\theta=0$ which I can calculate but can't understand it.
- Proof to sequences in real analysis
- The first homology group $ H_1(E(K); Z) $ of a knot exterior is an infinite cyclic group which is generated by the class of the meridian.
- Showing the Lie Algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are not isomorphic.
- Prove that $\int_0^{\infty} \frac{t^n} {1+e^t}dt=(1-2^{-n})n!\zeta (n+1)$