Notes_EN

Note to understand Galois theory

Notation In this article, we use following notation. $G$: Group, $R$: Ring, $F$: Field To be continued until we understand Galois theory Characteristic Prime ring The smallest subring of a ring is called Prime ring. In other words, prime r…

Some properties of a cyclic group

Let $G=\langle a\rangle$ be a cyclic group. Let $\eta\in\mathrm{End}\langle a\rangle$. Then "$\eta\in\mathrm{Aut}\langle a\rangle\Leftrightarrow \eta(a)$ is a generator of $\langle a\rangle$". Proof. If $\left |\langle a\rangle\right |=\in…

Some properties of a field

Properties of a (general) field $F$ Let $F$ be a field and $E/F$ a field extension. Commutative ring $R (\neq 0)$ is a field $\Leftrightarrow$ Ideals of $R$ are only $0\left(=(0)\right)$ or $R\left(=(1)\right)$ Proof. $(\Rightarrow):\,a\in…

Properties of a group

Some properties of a group $G$ Let $K\lhd G$, $K\subset H$ be a subgroup of $G$. 1 Map \begin{align} \begin{array}{ccc} \{H\,|\,K\subset H:\text{subgroup of } G\} &\rightarrow &\{\bar{H}\,|\bar{H}:\text{subgroup of } G/K\}\\ H&\mapsto&H/K …

How to rename files using script

Shell script Extract from 3rd to 5th characters of the file name Add prefix "P-" and suffix "-S" Batch file (For MS DOS) Ex. : Delete first two characters. (abcde.txt→cde.txt) Ex. : Add "prefix" at the beginning of a file name. (abc.txt→pr…

Determinant of ...

Determinant of $r^2\mathbb{1}-3\boldsymbol{r}\otimes\boldsymbol{r}$ \begin{align} \boldsymbol{r}\otimes\boldsymbol{r} =\begin{pmatrix} x^2&xy&xz\\ yx&y^2&yz\\ zx&zy&z^2 \end{pmatrix} \end{align} This matrix has following properties: $\math…

Easy way to derive rotation matrix

It is very easy to derive rotation matrix. This method is applied to general cases. Derivation Step1. Calculate rotation of unit vectors Step2. Arrange two vectors Background: Why can we derive? Derivation Step1. Calculate rotation of unit…

Useful knowledge for calculation

Prefix Pressure Others Density of water: Water pressure: Prefix Name Symbol $10^n$ kilo k $10^3$ hecto h $10^2$ deca da $10^1$ $10^0$ deci d $10^{-1}$ centi c $10^{-2}$ Pressure \begin{align} 1 \mathrm{\,Pa}&=1 \mathrm{\,N/m^2}\\ \text{Atm…

Contraction of completely anti-symmetric tensor

Contraction formula of the completely antisymmetric tensor (or Levi-Civita symbol) is indispensable for vector calculus.If you think it is hard to remember the formula, it is wrong. Actually, the derivation is very easy.At the end of this …