Fields

Definition: Field

A field $(F,+,\cdot )$ is an integral domain where each nonzero element has a multiplicative inverse, i.e. for each $a\in F,a\ne 0$ there exists an element $a^{-1}\in F$ such that

$$a\cdot a^{-1}=a^{-1}\cdot a=1$$

Theorem: Uniqueness of Multiplicative Inverses

The multiplicative inverse of an element $a$ in a field is unique.

PROOF

Suppose there were two multiplicative inverses $a^{-1}$ and $a^{-1}{}^{\prime }$.

Since $a^{-1}$ is a multiplicative inverse, we have

$$a\cdot a^{-1}=1$$

Similarly, since $a^{-1}{}^{\prime }$ is a multiplicative inverse, we have

$$a\cdot a^{-1}{}^{\prime }=1$$

Combining the two equations, we obtain

$$a\cdot a^{-1}=a\cdot a^{-1}{}^{\prime }$$

Multiply both sides by $a^{-1}$ - it does not matter whether we multiply from the left or from the right, since multiplication in integral domains is commutative.

$$a^{-1}\cdot (a\cdot a^{-1})=a^{-1}\cdot (a\cdot a^{-1}{}^{\prime })$$

We now avail ourselves of the associativity of multiplication.

$$(a^{-1}\cdot a)\cdot a^{-1}=(a^{-1}\cdot a)\cdot a^{-1}{}^{\prime }$$ $$1\cdot a^{-1}=1\cdot a^{-1}{}^{\prime }$$ $$a^{-1}=a^{-1}{}^{\prime }$$

Fields as Vector Spaces

Theorem: Fields as Vector Spaces

Every field $F$ with is a vector space $(F,F,+,\cdot )$ over itself.

PROOF

TODO

Ordered Fields

Definition: Ordered Field

An ordered field is a field $(F,+,\cdot )$ with a total order $\le$ which for all $a,b,c\in F$ satisfies

• If $a\le b$, then $a+c\le b+c$;

• If $0\le a$ and $0\le b$, then $0\le a\cdot b$.

NOTATION

We write $a<b$, if $a\le b$ and $a\ne b$.

The notations $b\ge a$ and $b>a$ stand for $a\le b$ and $a<b$, respectively.