A-3.2 Examples

$\mathbb{Z}$ (the set of all integers) is a ring, but not a field, because not all of its elements have a multiplicative inverse (as shown in §A-2.2).

$ℝ$ (the set of all real numbers) is a field. As shown in §A-2.2, it is an abelian group over the $\text{(+)}$ and $\text{(⋅)}$ operators, and its elements are distributive over the $\text{(+,⋅)}$ operators.

${\mathbb{Z}}_7=\{0,1,2,3,4,5,6\}$ is a finite field because:

• Closed: For any $a,b\in {\mathbb{Z}}_7$, there exists some $c_1\in {\mathbb{Z}}_7$ such that $a+b\equiv c_{1}mod7$ and, some $c_2\in {\mathbb{Z}}_7$ such that $a\cdot b\equiv c_{2}mod7$.
• Associative: For any $a,b,c\in {\mathbb{Z}}_7$, $(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
• Commutative: For any $a,b\in {\mathbb{Z}}_7$, $a+b=b+a$, and $a\cdot b=b\cdot a$.
• Distributive: For any $a,b,c\in {\mathbb{Z}}_7$, $(a+b)\cdot c=a\cdot c+b\cdot c$.
• Identity: For any $a\in {\mathbb{Z}}_7$, its additive identity is $0$, and its multiplicative identity is $1$.
• Inverse: For any $a\in {\mathbb{Z}}_7$, there exists an additive inverse $a_1^{\prime }\in {\mathbb{Z}}_7$ such that $a_1+a_1^{\prime }\equiv 0mod7$. For example, if $a_1=3$, then its additive inverse $a_1^{\prime }=4$, because $3+4=7\equiv 0mod7$. Also, for any $a_2\in {\mathbb{Z}}_7$ (except for 0), there exists a multiplicative inverse $a_2^{\prime }\in {\mathbb{Z}}_7$ such that $a_2\cdot a_2^{\prime }\equiv 1mod7$. For example, if $a_2=3$, then its multiplicative inverse $a_2^{\prime }=5$, because $3\cdot 5=15\equiv 1mod7$.