Intereting Posts

Does the open mapping theorem imply the Baire category theorem?
Spectral Measures: Scale Embeddings
Is the set of quaternions $\mathbb{H}$ algebraically closed?
Calculating probability of 'at least one event occurring'
Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point
If $\sin A + \cos A + \tan A + \cot A + \sec A + \csc A = 7$ then $x^2 – 44x – 36 = 0$ holds for $x=\sin 2A$
Closed form of $\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx$
Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$
Is there a initial “bordism-like” homology theory?
If $p$ is a prime integer, prove that $p$ is a divisor of $\binom p i$ for $0 < i < p$
Volume of Pyramid
Numerical integration over a surface of a sphere
How can one prove that $\pi^4 + \pi^5 < e^6$?
Proof that every finite dimensional normed vector space is complete
What is a conormal vector to a domain intuitively?

Prove that the set of symbols $\{a+bi \mid a, b \in \mathbb{F}_3\}$ forms a field with nine elements, if the laws of composition are made to mimic addition and multiplication of complex numbers. Will the same method work for $\mathbb{F}_5$? For $\mathbb{F}_7$?

I was able to prove that $\{a+bi \mid a, b \in \mathbb{F}_3\}$ is a field and that the method does not work for $\mathbb{F}_5$. But could someone explain to me why it works in $\mathbb{F}_7$ ?

Thank you

- Irreducibility of $x^{n}+x+1$
- Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible
- What is the condition for a field to make the degree of its algebraic closure over it infinite?
- Let $F$ be a field in which we have elements satisfying $a^2+b^2+c^2 = −1$. Show that there exist elements satisfying $d^2+e^2 = −1$.
- How does $\cos(2\pi/257)$ look like in real radicals?
- For $f\in\mathbb{Q}$, Gal($f)\subset S_n$ is a subset of $A_n$ iff $\Delta(f)$ is a square in $\mathbb{Q}^*$

- Is there a subfield $F$ of $\Bbb R$ such that there is an embedding $F(x) \hookrightarrow F$?
- Subfield of rational function fields
- Intermediate fields of a finite field extension that is not separable
- Show from the axioms: Addition in a quasifield is abelian
- Field reductions
- Why must a field with a cyclic group of units be finite?
- Field extensions and algebraic/transcendental elements
- Characteristic of a field is $0$ or prime
- Show that $\mathbb{Q}(\sqrt{2}+\sqrt{5})=\mathbb{Q}(\sqrt{2},\sqrt{5})$
- Let $F$ be a field of characteristic $p$ and let $f(x)=x^p-a\in F.$ Show that $f(x)$ is irreducible over $F$ or $f(x)$ splits in $F.$

Everything is completely straightforward over $\mathbb{F}_p$, for any prime $p$, with the possible exception of showing that every nonzero element has a multiplicative inverse.

If you remember how to express reciprocals $\frac1{a + bi}$ by “rationalizing the denominator” (multiply numerator and denominator by the conjugate $a – bi$), then the idea is that this should work here too, as long as $a^2 + b^2$ is guaranteed to be non-zero (as long as one of $a, b$ is nonzero modulo $7$). So you have to show that $a^2 +b^2 = 0$ has no solutions modulo $7$ besides $a = 0 = b$ (modulo $7$, of course). Can you show this is equivalent to $-1$ being a nonsquare modulo $7$?

- Moments of Particular System of Stochastic Differential Equations (SDEs)
- How can I prove that this group is isomorphic to a semidirect product?
- Why is homology invariant under deformation retraction/homotopy equivalence?
- Using $\epsilon$-$\delta$ definition to prove that $\lim_{x\to-2}\frac{x-1}{x+1}=3$.
- About stationary and wide-sense stationary processes
- Why can ALL quadratic equations be solved by the quadratic formula?
- Showing $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{R},+)$
- Distributions on manifolds
- Why do $n$ linearly independent vectors span $\mathbb{R}^{n}$?
- Show that $S$ is a group if and only if $aS=S=Sa$.
- Prove that $2730$ divides $n^{13} – n$ for all integers $n$.
- Proving two integral inequalities
- Prove $\{(x,y) \in \mathbb R^2|x^2 + y^2 > 1 \}$ is not simply connected
- Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?
- How many solutions does the equation $x_1 + x_2 + x_3 = 14$ have, where $x_1$ , $x_2$ , $x_3$ are non-negative integers.