Notation
In this article, we use following notation.
- $G$: Group, $R$: Ring, $F$: Field
To be continued until we understand Galois theory
Characteristic
In other words, prime ring is a subring generated by $1$.
Consider the following map from $\mathbb{Z}$ to a ring $R$,
\begin{align}
n\mapsto 1_{R}.
\end{align}
This map is a homomorphism and the image is the prime ring of $R$.
Similarly, prime field of a field $F$ is defined as the smallest subfield of $F$.
- If characteristic of $F$ is $p\neq 0$, $\text{"prime ring of $F$"}\cong\mathbb{Z}/(p)$ is a subfield.
(Note that the prime ring of a domain has characteristic $0$ or a prime $p$.) - If characteristic of $F$ is $0$, $\text{"Prime field of $F$"}\cong \mathbb{Q}$.
(Since the monomorphism $\mathbb{Z}\ni m\mapsto m1\in \text{"Prime field of $F$"}$ can be extended to a monomorphism $\mathbb{Q}\ni n/m\mapsto n1(m1)^{-1}\in \text{"Prime field of $F$"}$.)
An extension field $E/F$ is called a splitting field over $F$ of $f(x)$ if the following conditions are satisfied:
- $E=F(r_1,...,r_n)$
- $f(x)=(x-r_1)\cdots (x-r_n)$
Let $S$ be a generator of a ring $R$ (or a field $F$) .
If two homomorphism $\eta_i:R\rightarrow R^\prime\,(i=1,2)$ are equal on $S$, $\eta_1=\eta_2$.
This is because $S\subset\bigl\{r\in R\bigl| \eta_1(r)=\eta_2(r)\bigr\}$ is a ring (field).
Let $\eta:R\rightarrow R^\prime$ be a homomorphism of commutative ring and $u\in R^\prime$.
Then we get a unique extension $\eta_u:R[x]\rightarrow R^\prime$ such that
\begin{align}
\eta_u(x)=u.
\end{align}
Field isomorphism $\eta:F\rightarrow \bar{F}$ has a unique extension $\tilde{\eta}:F[x]\rightarrow \bar{F}[x]$ such that
\begin{align}
\tilde{\eta}(x)=x.
\end{align}
We write $\bar{g}(x):=\tilde{\eta}\bigl(g(x)\bigr)$ below.
Our (First) Goal ~to be contained above
$\Leftrightarrow$ Galois group is solvable