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…

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 …

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 $r^2\mathbb{1}-3\boldsymbol{r}\otimes\boldsymbol{r}$ \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…