## Algebra

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…

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…

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 …