Intereting Posts

Why does Friedberg say that the role of the determinant is less central than in former times?
Is $\mathbb Q_r$ algebraically isomorphic to $\mathbb Q_s$ while r and s denote different primes?
Example of a function continuous at only one point.
Unbiased Estimator for a Uniform Variable Support
Diophantine quartic equation in four variables, part deux
$L^p$ and $L^q$ space inclusion
Variance of the random sum of a Poisson?
Number of solutions of equation
Proving the sum of two independent Cauchy Random Variables is Cauchy
how to prove that every low-dimensional topological manifold has a unique smooth structure up to diffeomorphism?
Lattice of continuous functions
Fourier transform in $L^p$
How to show a level set isn't a regular submanifold
Help on a proof about Ramsey ultrafilters
How many solutions possible for the equation $x_1+x_2+x_3+x_4+x_5=55$ if

For example, over fields with characteristic 2, there exist nonzero symmetric nilpotent matrices, and nonzero matrices could be simultaneously symmetric and anti-symmetric. I wonder why characteristic 2 makes such fields so special.

- Why any field is a principal ideal domain?
- Irreducible but not prime in $\mathbb{Z} $
- Prove that $k/(xy-zw)$, the coordinate ring of $V(xy-zw) \subset \mathbb{A}^4$, is not a unique factorization domain
- When do we use Tensor?
- Find the number of irreducible polynomials in any given degree
- Some irreducible character separates elements in different conjugacy classes
- The ring $\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$
- How is a group made up of simple groups?
- Schröder-Bernstein for abelian groups with direct summands
- Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$

All fields of nonzero characteristic are ‘pathological’ in some sense. It’s just easier to trip over a problem with $2$ than a problem with, say, $1319$.

Symmetric nilpotents exist in all characteristics. For example, in characteristic 3, you have

$$ \left( \begin{matrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right) $$

as an example of a matrix that squares to zero. It’s easy to generalize this to any positive characteristic.

Squaring behaves strangely in characteristic 2. Among the oddities is that there is only one square root of 1. In some sense, this is responsible for the thing with symmetric and anti-symmetric.

In characteristic 3, it’s cubing that’s strange, and so forth.

Mostly, it’s because $a=-a$ in fields of characteristic $2$.

- Applications of conformal mapping
- $\mathbb{Q}(\sqrt{1-\sqrt{2}})$ is Galois over $\mathbb{Q}$
- Simple question about the definition of Brownian motion
- A question about an $n$-dimensional subspace of $\mathbb{F}^{S}$.
- Is there a non-trivial countably transitive linear order?
- Prove that the union of three subspaces of $V$ is a subspace iff one of the subspaces contains the other two.
- Infinite subset with pairwise comprime elements
- General solution for the Eikonal equation $| \nabla u|^2=1$
- showing $\psi: R\to \mathbb C$ is ring isomorphism.
- Proof of elliptic curves being an abelian group
- Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges
- Sum of Cauchy distributed random variables
- $\mathbb Z^n/\langle (a,…,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$
- Weak Convergence of Positive Part
- Can $\{(f(t),g(t)) \mid t\in \}$ cover the entire square $ \times $ ?