Intereting Posts

Prove. Let {v1,v2,v3} be a basis for a vector space V. Show that {u1,u2,u3} is also a basis, where u1=v1, u2=v1 +v2, and u3=v1+v2+v3
Dominated convergence for sequences with two parameters, i.e. of the form $f_{m,n}$
What is $\arctan(x) + \arctan(y)$
If $H$ is a proper subgroup of a $p$-group $G$, then $H$ is proper in $N_G(H)$.
Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$
How do I transform the equation based on this condition?
Generalized Fresnel integral $\int_0^\infty \sin x^p \, {\rm d}x$
No closed form for the partial sum of ${n\choose k}$ for $k \le K$?
Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Comment on S.
What is the Fourier transform for $f(x)=e^{-x^2}$
Limit of $a_n = \sum\limits_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$
Prove that $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}$
Set of all injective functions $A\to A$
What is the remainder when $40!$ is divided by $1763$?
How to calculate gradient of $x^TAx$

For any prime $p$ find the number of monic irreducible polynomials of degree $2$ over

$\mathbb Z_p$. Do the same problem for degree $3$. Generalize the above statement to higher degree polynomials as much as you can.

My idea for degree 2:

assume that polynomial is reducible, then we can write into this form: $(x-m)(x-n)=0$, expand this, so $x^2-(m+n)x+mn=0$,we can use a matrix to capture all possible value of $(m+n) and $ $mn$, like when $p=3$, $m+n$ matrix is $\begin{matrix}

0&1&2\\

1&2&0\\

2&0&1\end{matrix}$, similarly, we can write mn, find the different value, that m,n must be irreducible when we expand (x-m)(x-n)=0.

- Isomorphisms: preserve structure, operation, or order?
- Group isomorphism concerning free group generated by $3$ elements.
- Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2$
- $F/(x^2)\cong F/(x^2 - 1)$ if and only if F has characteristic 2
- if $\Omega=\{1,2,3,\cdots \}$ then $S_{\Omega}$ is an infinite group
- Intermediate fields between $\mathbb{Z}_2 (\sqrt{x},\sqrt{y})$ and $\mathbb{Z}_2 (x,y)$

- What's the motivation of the definition of primary ideals?
- How many ways can 10 teachers be divided among 5 schools?
- Minimum number of out-shuffles required to get back to the start in a pack of $2n$ cards?
- How to classify one-dimensional F-algebras?
- What is the probability that when you place 8 towers on a chess-board, none of them can beat the other.
- Axiomatic approach to polynomials?
- Different ways of Arranging balls in boxes
- Are all simple left modules over a simple left artinian ring isomorphic?
- If I randomly generate a string of length N from an alphabet {A, B, C}, what's the likelihood that exactly k characters will be the same?
- Irreducibility of $x^p-x-c$

There are $p^2$ total monic polynomials of degree 2 (where the $p^2$ counts the possible linear and constant term combinations). There are $p+\binom{p}{2}$ monic polynomials of degree 2 that are reducible (where the $p$ counts the ones with a repeated root, and the $\binom{p}{2}$ counts the ones with two distinct roots). So there are $$p^2-\left(p+\binom{p}{2}\right)=\frac{p^2-p}{2}$$ irreducible quadratics. Can you extend this approach to cubics?

- Minkowski Inequality for $p \le 1$
- $AB=BA$ with same eigenvector matrix
- When do weak and original topology coincide?
- the composition of $g \circ f$ is convex on $\Omega$ from Shapley's definition of convexity
- Cauchy-Schwarz for integrals
- Unique Stationary Distribution for Reducible Markov Chain
- Math story: Ten marriage candidates and 'greatest of all time'
- How to show pre-compactness in Holder space?
- Is my understanding of antisymmetric and symmetric relations correct?
- Evaluating $\int_{0}^{1} \sqrt{1+x^2} \text{ dx}$
- How to prove that Gödel's Incompleteness Theorems apply to ZFC?
- Poincaré's theorem about groups
- How to solve the integral $\int\frac{x^2-1}{(x^2+1)\sqrt{x^4+1}}\,dx$ .
- Why isn't the directional derivative generally scaled down to the unit vector?
- How to find an ellipse , given 2 passing points and the tangents at them?