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 under $\text{(+)}$; its nonzero elements form an abelian group under $\text{(⋅)}$, and multiplication distributes over addition.

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

• Closed: For any $a,b\in {\mathbb{Z}}_7$, there exist $c_1,c_2\in {\mathbb{Z}}_7$ such that $a+b\equiv c_1(\mathit{𝑚𝑜𝑑}7)$ and $a\cdot b\equiv c_2(\mathit{𝑚𝑜𝑑}7)$.
• 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 $a,b,c\in {\mathbb{Z}}_7$, $(a+b)\cdot c=a\cdot c+b\cdot c$, and $a\cdot (b+c)=a\cdot b+a\cdot c$.
• Identity: The additive identity is $0$, and the multiplicative identity is $1$.
• Inverse: For any $a\in {\mathbb{Z}}_7$, there exists $a^{\prime }\in {\mathbb{Z}}_7$ such that $a+a^{\prime }\equiv 0(\mathit{𝑚𝑜𝑑}7)$ (e.g., the additive inverse of $3$ is $4$ since $3+4\equiv 0(\mathit{𝑚𝑜𝑑}7)$). For any $a\in {\mathbb{Z}}_7^{\times }=\{1,\dots ,6\}$, there exists $b\in {\mathbb{Z}}_7^{\times }$ such that $\mathit{𝑎𝑏}\equiv 1(\mathit{𝑚𝑜𝑑}7)$ (e.g., $3\cdot 5=15\equiv 1(\mathit{𝑚𝑜𝑑}7)$).