Intereting Posts

What is the integral of $e^{a \cdot x+b \cdot y}$ evaluated over the Koch Curve
Direct proof that $n!$ divides $(n+1)(n+2)\cdots(2n)$
Finding $\sum\limits_{n=2}^\infty \ln\left(1-\frac{1}{n^2}\right)$
How to prove that $\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x)$
What does “sets of arbitrarily large measure” mean — question about $L_p$ embeddings
Is $\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$?
Do there exist closed subspaces $X$, $Y$ of Banach space, such that $X+Y$ is not closed?
In the card came “Projective Set”, show that 7 cards do always contain a set.
On Shanks' quartic approximation $\pi \approx \frac{6}{\sqrt{3502}}\ln(2u)$
Minimum number of points chosen from an $N$ by $N$ grid to guarantee a rectangle?
Difficulty in understanding a part in a proof from Stein and Shakarchi Fourier Analysis book.
Find a basis for $U+W$ and $U\cap W$
Application of Banach-Steinhaus theorem
Prove a $\pi$ inequality: $\left(1+\frac1\pi\right)^{\pi+1}<\pi$
Find a Solution using Green's Function

Let $K=F_2[x]/(x^3+x+1)$. I want to show that $f(x)=x^4+x^2+1$ is reducible over $K$ but has no roots in it. How to proceed? I know that $F$ contains 8 elements, how is the structure of these elements? How can I realize the elements of this field?

- Finding the degree of a field extension over the rationals
- Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$
- Is the algebraic subextension of a finitely generated field extension finitely generated?
- Are there broad or powerful theorems of rings that do not involve the familiar numerical operations (+) and (*) in some fundamental way?
- Why should I care about fields of positive characteristic?
- Rigidity of the category of fields
- Sum and Product of two transcendental numbers cannot be simultaneously algebraic
- Isomorphisms: preserve structure, operation, or order?
- If $F$ is a formally real field then is $F(\alpha)$ formally real?
- If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field.

**Hint:** Find all quadratic polynomials over $\mathbb{F}_{2}$ and find

two of which that when multiplied, and using $x^{3}+x+1\equiv0$,

gives you $f$.

- Trying to Prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a Group of Infinite Order
- Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.
- Proving Cauchy condensation test
- When is $\mathbb{Z}$ not an UFD (for $d>1$)?
- Using Octave to solve systems of two non-linear ODEs
- Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
- Does $\int_{1}^{\infty}\sin(x\log x)dx $ converge?
- Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
- Understanding common knowledge in logic and game theory
- Functions satisfying $f(m+f(n)) = f(m) + n$
- Ideal class group of $\mathbb{Q}(\sqrt{-103})$
- Are “$n$ by $n$ matrices with rank $k$” an affine algebraic variety?
- Variable leaving basis in linear programming – when does it happen?
- How do we know what the integral of $\sin (x)$ is?
- How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?