Field

A field is a set paired with two operations on the set, which are designated as addition ($+$) and multiplication ($\cdot$). As a group can be conceptualized as an ordered pair of a set and an operation, $\left(G, \cdot \right)$, a field can be conceptualized as an ordered triple $\left(F,+,\cdot \right)$.

A set with addition and multiplication, $\left(F,+,\cdot \right)$, is a field if and only if it satisfies the following properties:

1. Commutativity of both addition and multiplication — For all $a, b \in F$, $a + b = b + a$ and $a\cdot b=b\cdot a$;
2. Associativity of both addition and multiplication — For all $a,b,c \in G$, $\left(a+b\right)+c=a+\left(b+c\right)$ and $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$;
3. Additive Identity — There exists a "zero" element, $0\in F$, called an additive identity, such that $a+0$, for all $a \in F$;
4. Additive Inverses — For each $a \in F$, there exists a $b\in F$, called an additive inverse of $a$, such that $a+b=0$;
5. Multiplicative Identity — There exists a "one" element, $1\in F$, different from 0, called a multiplicative identity, such that $1\cdot a=a$, for all $a \in F$;
6. Multiplicative Inverses — For each $a \in F$, except for 0, there exists a $c\in F$, called a multiplicative inverse of $a$, such that $a\cdot c=1$;
7. Distributive property — For all $a, b, c \in F$, $a\cdot \left(b+c\right)=a\cdot b+a\cdot c$;
8. Closure of addition and multiplication — For all $a, b \in F$, $a+b\in G$ and $a \cdot b \in G$.

We will often abbreviate the multiplication of two elements, $a\cdot b$, by juxtaposition of the elements, $ab$. Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, $a + bc + d = a + \left(bc\right) + d$.

We can also denote $-a$ and $a^{-1}$ as additive and multiplicative inverses of any $a \in F$. Furthermore, we can define two more operations, called subtraction and division by $a - b = a + \left(-b\right)$, and provided that $b\neq 0$, $\frac{a}{b}=a\cdot b^{-1}$.

Important Results

From the given criterion for a field, it can be shown that:

• $\left(F,+\right)$ and $\left(F^{*},\cdot \right)$ are abelian groups (where $F^*$ is the set $F$ excluding 0), and therefore, group-theoretic properties can be applied to fields, such as uniqueness of identities and inverses, and cancellation properties;
• $0\cdot a=0$, for all $a \in F$;
• $a\left(-b\right) = \left(-a\right)b = -\left(ab\right)$, for all $a, b \in F$;
• $\left(-a\right)\left(-b\right)=ab$, for all $a, b \in F$;
• $\left(-1\right)\cdot a=-a$, for all $a \in F$;
• $\left(-1\right)\cdot \left(-1\right)=1$, for all $a \in F$;
• Multiplication distributes over subtraction.

Optional Properties

A field $\left(F,+,\cdot \right)$ is:

• A subfield of a field $\left(K,+,\cdot \right)$ if $F\subseteq K$ (see subset), where addition and multiplication on $F$ is a domain restriction on the addition and multiplication on $K$. More commonly, we say that $K$ is an extension field of $F$;
• An ordered field if there exists a total order $\le$ on $F$ such that for all $a,b,c \in G$, if $a \le b$, then $a + c \le b + c$, (translation invariance), and if $0\le a$ and $0 \le b$, then $0\le ab$.