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 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 $R$.

Characteristic of a ring
\begin{align} \text{Prime ring} \cong \begin{cases} \,\mathbb{Z}/(0)=\mathbb{Z} &\text{: Called characteristic $0$}\\ \,\mathbb{Z}/(k) \quad \text{for some } k>0 &\text{: Called characteristic $k$} \end{cases} \end{align}

Similarly, prime field of a field $F$ is defined as the smallest subfield of $F$.

  1. 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 $p$.)
  2. 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 $F$"}$.)
Splitting Field
Let $f(x)\in F[x]$ be a monic polynomial, that is, the coefficient of the highest degree term is $1$.
An extension field $E/F$ is called a splitting field over $F$ of $f(x)$ if the following conditions are satisfied:
  1. $E=F(r_1,...,r_n)$
  2. $f(x)=(x-r_1)\cdots (x-r_n)$
Existence of a splitting field
Any monic polynomial $f(x)\in F[x]$ of positive degree has a splitting field $E/F$.
Proof.

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.

Galois group
Galois group of $E$ over $F$ is defined by \begin{align} \mathrm{Gal\,} (E/F) :=\mathrm{Aut\,} (E/F) =\bigl\{\eta\,\bigl|\, \eta\text{: isomorphism of $E$ to $E$, }\eta|_F=\mathrm{id}_F\bigr\}. \end{align}
The subfield of $G$-invariants
Let $E$ be a field and $G$ be a subgroup of $\mathrm{Aut\,}E$. We call \begin{align} \mathrm{Inv\,}G:=\bigl\{a\in E\,\bigl|\, \eta(a)=a\quad(\forall \eta\in G)\bigr\} \end{align} the subfield of $G$-invariants or the $G$-fixed subfield of $E$.
Equation solvable by radicals
Let $f(x)\in F[x]$ be a monic of positive degree. Equation $f(x)=0$ is said to be solvable by radicals on $F$ if there exists $K/F$ such that \begin{align} &F=F_1\subset F_2\subset\cdots\subset F_{r+1}=K\quad\text{(: Called "Root tower over $F$")}\\ &\bigl(F_{i+1}=F_i(d_i), \quad d_i^{n_i}\in F_i\bigr) \end{align} and $K$ contains a splitting field over $F$ of $f(x)$.

Our (First) Goal ~to be contained above

Galois Critarion
Equation $\bigl(F[x]\ni\bigr) f(x)=0$ is solvable by radicals over $F$ of characteristic $0$
$\Leftrightarrow$ Galois group is solvable

Reference

Basic Algebra I: Second Edition (Dover Books on Mathematics)