Uniqueness Field Properties

Properties

1. Additive Inverse is Unique
2. Multiplicative Identity is Unique
3. Additive Inverse is Unique
4. Multiplicative Inverse is Unique
5. Addition is Cancellational $(\forall a,b\in F,a+b=a+c⟹b=c)$
6. If $a\ne 0$, Multiplication is Cancellational $(\forall a,b,c\in \mathbb{F},a\ne 0,a\ast b=a\ast c⟹b=c)$
7. Addition Distributes nicely ($(a+b)c=ac+bc$)
8. If $a,b\in \mathbb{F}$, then if $a\times 0=0⟹a=0\vee b=0$

Examples

Showing $Z_6$ is not a Field

Consider

• $3\times 5\mathrm{mod}6=15\mathrm{mod}6=3$
• $3\times 1\mathrm{mod}6=3\mathrm{mod}6=3$
• Then, we have more than one element acts as the Multiplicative Identity. Thus, $Z_6$ is not a field!

Showing $Z_p$ where $p$ is Prime Number is Field

Proving $⟹$

Proof By Contra-Positive.

• Suppose $p$ is not prime, then $p$ is a Composite Number. by defn of Congruent Modulo as $n\ge 2$
• Thus, $p=ab$ where $ab\in \{0,\dots ,p-1\}$, $ab\mathrm{mod}p=0$, but $a\ne 0,b\ne 0$ so ${\mathbb{Z}}_p$ fails the last property

Proving $⟸$

• Let $a\in {\mathbb{Z}}_p$ s.t $a\ne 0$, then the GCD $gcd(a,p)=1$ by defn of prime number, and as $a<p$.
• Then, $S=\{1{\times }_{\mathrm{mod}p}a,2{\times }_{\mathrm{mod}p}a,\dots ,(p-1){\times }_{\mathrm{mod}p}a\}$
• If we show this set contains 1, we show that a Multiplicative Inverse must always exist. We show that $S$ does not contain $0$, and $S$ does not contain a repeated element
• Showing $0\notin S$
  • As $p$ is prime, then $\nexists b\in p\text{ s.t }b\ast a=np$
• Now, we show that $j{\times }_{\mathrm{mod}p}a\ne i{\times }_{\mathrm{mod}p}a$ if $i\ne j$