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…

# 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…

# 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 …