[TOC]

0.1. Def(field)

A field is a set $F$ together with two operations

addition:$F\times F\to F:(a,b)\mapsto a+b$ and

multiplication:$F\times F\to F:(a,b)\mapsto a\cdot b$

satisfying the following properties for $\forall a,b,c\in F$

• Associativity of addition and multiplication:

$a+(b+c)=(a+b)+c$ and $a\cdot (b\cdot c)=(a\cdot b)\cdot c$.

• Commutativity of addition and multiplication:

$a+b=b+a$ and $a\cdot b=b\cdot a$

• Additive and multicative identity:

$\exists 0,1\in F$ such that $a+0=a$ and $a\cdot 1=a$.

• Additive inverses:

$\forall a\in F,\exists -a\in F$, called the additive inverse of $a$, such that $a+(-a)=0$.

• Multiplicative inverses:

$\forall a\in F\text{with}a\ne 0,\exists a^{-1}\in F$, called multiplicative inverse of $a$ such that $a\cdot a^{-1}=1$.

• Distibutivity of multiplication over addition:

$a\cdot (b+c)=(a\cdot b)+(a\cdot c)$.

Note:

• $a+b$ is called the sum of $a$ and $b$.
• $a\cdot b$ is called the product of $a$ and $b$.

上記の定義から以下は簡単に示せる(証明略)

$a\cdot 0=0,-a=(-1)\cdot a\forall a\in A$

$ab=0\Rightarrow a=0\vee b=0\forall a,b\in A$

$(-1)\cdot (-1)=1$ (proof is below)

$$\begin{array}{cc}0& =0\cdot 0\\ & =(1-1)\cdot (1-1)\\ & =1\cdot 1+(-1)\cdot 1+1\cdot (-1)+(-1)\cdot (-1)\\ & =1+(-1)+(-1)+(-1)\cdot (-1)\end{array}$$

0.2. Def(ordered_field)

A field $(F,+,\cdot )$ together with a (strict) total order) $<$ on $F$ is an ordered field if the order satisfies the following properties for all $a,b$ and $c$ in $F$;

• If $a<b$ then $a+c<b+c$, and
• If $0<a$ and $0<b$ then $0<a\cdot b$.

Property

• 1 is positive (proof))

0.3. Th(Square_is_positive)

Let $(F,+,\cdot )$ together with a (strict) total order) $<$ on $F$ be an ordered field.

Then $x\in F\setminus \left\{0\right\}\Rightarrow x^2>0$

Therefore $1=1\times 1>0$

(Proof)$x\in F\setminus \left\{0\right\}$とすると$x>0$もしくは$x<0$が成立.$x>0$の場合はordered fieldの定義より成立.$x<0$の場合は両辺に$x$の逆元を足すことで$0<-x$となる.この時fieldの性質)より$0<(-x)\cdot (-x)=(-1)\cdot x\cdot (-1)\cdot x=(-1)\cdot (-1)\cdot x\cdot x=x^2$

0.4. Def(absolute_value)

A real-valued function $v$ on a field $F$ is called an absolute value if it satisfied the following four axioms.

• Non-negativity

  $v\left(a\right)\ge 0$

• Positive-definiteness

  $v\left(a\right)=0\iff a=0$

• Multiplicativity

  $v\left(ab\right)=v\left(a\right)v\left(b\right)$

• Subadditivity or the triangle inequality

  $v(a+b)\le v\left(a\right)+v\left(b\right)$

Where $0$ donotes the additive identity element of $F$.

• It follows from positive-definiteness and multiplicativity that $v\left(1\right)=1$, where $1$ denotes the multiplicative identity element of $F$(証明略).
• $v(-a)=v\left(a\right)$ for every $a\in F$(下記に証明)

If $v$ is an absolute value on $F$, then the function $d$ on $F\times F$, defined by $d(a,b)\equiv v(a-b)$, is a metric(証明略).

$v(-a)=v\left(a\right)$ for every $a\in F$であることを示す.

$1=v\left(1\right)=v(-1\cdot -1)=v(-1)v(-1)$

であるので,$v(-1)=1$であることに注意すると,任意の$a\in F$に対して$v(-a)=v(-1\cdot a)=v(-1)v\left(a\right)=v\left(a\right)$

0.5. Def(valued_field)

$(K,|\cdot |)$ is called a valued field if $K$ is a field and $|\cdot |:K\to R$ is an absolute value).

0.6. Def(group)

A group is a noempty set $G$ on which there is definied a binary operation $(a,b)\to ab$ satisfying the following properties:

Closure:If $a$ and $b$ belong to $G$, then $ab$ is also in $G$;

Associativity:$a\left(bc\right)=\left(ab\right)c$ for all $a,b,c\in G$;

Identity: There is an element $1\in G$ such that $a1=1a=a$ for all $a$ in $G$;

Inverse:If $a$ is in $G$, then there is an element $a^{-1}$ in $G$ such that $a{a}^{-1}=a^{-1}a=1$.

A abelian group (commutative group) is a group satisfying the following properties;

Commutativity:$ab=ba$.

• binary operationのことをgroup law of $G$ということもある
• $ab$を$a\cdot b$と表記することもある

Reference

0.7. Def(Additive_group)

An additive group is a group of which the group operation is to be thought of as addition in some sense. It is usually abelian, and typically written using the symbol + for its binary operation.

0.8. Def(group_action)

If $G$ is a group and $X$ is a set, then a (left) group action $\varphi$ of $G$ on $X$ is a function

$\varphi :G\times X\to X$:$(g,x)\mapsto \varphi (g,x)$

that satisfies the following two axioms (where we denote $\varphi (g,x)$ as $g\cdot x$):

Identity:$e\cdot x=x$ for all $x$ in $X$.

where $e$ denotes the identity element of the group $G$.

Compatibility:$\left(gh\right)\cdot x=g\cdot (h\cdot x)$ for all $g,h$ in $G$ and all $x$ in $X$.

The group $G$ is said to act on $X$(on the left). The set $X$ is called a (left) G-set.

A group action is transitive if $X$ is non-empty and if for each pair $x,y$ in $X$ there exists a $g$ in $G$ such that $g\cdot x=y$.

A group action is faithful (or effective) (1) if for every two distinct $g,h$ in $G$ there exists an $x$ in $X$ such that $g\cdot x\ne h\cdot x$; or equivalently, (2) if for each $g\ne e$ in $G$ there exists an $x$ in $X$ such that $g\cdot x\ne x$.

A group action is free (or semi regular or fixed point free) if,(3) given $g,h$ in $G$, the existance of an $x$ in $X$ with $g\cdot x=h\cdot x$ implies $g=h$.

Equivalently: (4) If $g$ is a group element and there exists an $x$ in $X$ with $g\cdot x=x$ (that is, if $g$ has at least one fixed point), then $g$ is the identity.

Note that (5) a fee action on a non-empty set is faithful.

(5)は簡単に示せるので略.その他も定義通りだが一応証明を載せる.

Proof (2) $\iff$ (1)

逆は自明なので,(2) $\Rightarrow$ (1)のみを示す.互いに異なる$g,h\in G$を任意にとる.この時$h^{-1}g=e\iff g=h$であるので,$h^{-1}g\ne e$である.よって,(2)より$x\in X$が存在し,$h^{-1}g\cdot x\ne x$であるので$g\cdot x\ne h\cdot x$.

Proof (3) $\Rightarrow$ (4)

任意の$g\in G$ に対して$x\in X$が存在し,$g\cdot x=x=e\cdot x$が成立しているとする.この時(1)の仮定より$g=e$, つまり$g$はidentity.

Proof (4) $\Rightarrow$ (3)

任意の$g,h\in G$に対して$x\in X$が存在し,$g\cdot x=h\cdot x$が成立しているとする.

この時 $$g\cdot x=h\cdot x\iff h^{-1}\cdot (g\cdot x)=h^{-1}\cdot (h\cdot x)\iff \left(h^{-1}g\right)x=x$$ であるので,$h^{-1}g=e\iff g=h$となる.