Intereting Posts

Limit evaluate $\lim_{x\to0}{{\frac{\ln(\cos(4x))}{\ln(\cos(3x))}}}$?
$2\times2$ matrices are not big enough
On the relationship between the commutators of a Lie group and its Lie algebra
I'm looking for some mathematics that will challenge me as a year $12$ student.
When is $\binom{n}{k}$ divisible by $n$?
Are these numbers $h_{r,s}$ irrational?
In how many ways can a $31$ member management be selected from $40$ men and $40$ women so that there is a majority of women?
Assumption that $\delta q'$ is small in the derivation of Euler-Lagrange equations.
Every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments
How is it solved: $\sin(x) + \sin(3x) + \sin(5x) +\dotsb + \sin(2n – 1)x =$
Testing the series $\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}$
What are the epimorphisms in the category of Hausdorff spaces?
Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$
Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
Prove by mathematical induction that $n^3 – n$ is divisible by $3$ for all natural number $n$

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$?

I’m trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} \equiv {a \choose b}.$ To do this I use FLT to show that $(1+x)^{pa} \equiv (1+x)^a \mod p$ and compare the coefficients of $x^b$ to complete the proof. Am I applying FLT correctly?

In general, do most theorems regarding integers/reals generalize to polynomials over the integers/reals? Are there some common pitfalls that I could make when trying to generalize such theorems?

- The units of $\mathbb Z$
- Intersection maximal ideals of a polynomial ring
- Generating Sets for Subgroups of $(\Bbb Z^n,+)$.
- Intuitive definitions of the Orbit and the Stabilizer
- Does every Abelian group admit a ring structure?
- What does it mean here to describe the structure of this quotient module?

- Prove that g has no roots
- How to simplify a square root
- Exact sequences of $SU(N)$ and $SO(N)$
- Real-world uses of Algebraic Structures
- Without the Axiom of Choice, does every infinite field contain a countably infinite subfield?
- Tips for finding the Galois Group of a given polynomial
- Ring with convolution product
- Why “characteristic zero” and not “infinite characteristic”?
- Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$
- Order of products of elements in a finite Abelian group

The answer to the first question is **NO**. When you expand $g(x)$ by the binomial theorem, you will see that all of the coefficients, except for the first and last, are zero, so $g(x)$ is $1+x^p$ modulo $p.$ The difference between $f(x)$ and $g(x)$ has $p$ roots in the algebraic closure of $\mathbb{F}_p,$ which is not the same as the number of roots of the zero polynomial.

I think that gives an indication of the problems for your second question.

Possibly you have encountered a common stumbling point: confusing polynomial *functions* with *formal* polynomials. Over $\,\Bbb F_p,\,$ by $\,\mu$Fermat, $\,f-g\, =\, x^p -x \,$ equals the constant *function* $\,0,\,$ but it is not equal to $\,0\,$ as a *formal* polynomial since, by definition, formal polyomials are equal iff their corresponding coefficients are equal.

- Given a local diffeomorphism $f: N \to M$ with $M$ orientable, then $N$ is orientable.
- Integration with respect to counting measure.
- Visualizing a homotopy pull back
- A normal subgroup intersects the center of the $p$-group nontrivially
- Probability of $5$ fair coin flips having strictly more heads than $4$ fair coin flips
- The Matrix Equation $X^{2}=C$
- The Duals of $l^\infty$ and $L^{\infty}$
- Is it possible that all subseries converge to irrationals?
- What is the rank of the cofactor matrix of a given matrix?
- Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?
- A question about sigma-algebras and generators
- Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$
- sum of irrational numbers – are there nontrivial examples?
- $g(x) = 1/(1+x^2)$ is continuous everywhere epsilon delta approach
- Group with order $p^2$ must be abelian . How to prove that?