Field

A field $(F, +, ·)$ is a set $F$ with two maps $+ : F × F → F$ and $· : F × F → F$ called addition and multiplication, such that the following conditions (field axioms) hold:

1. $(F, +)$ is a commutative group, where the neutral element is denoted by $0$.

   1.1 Associativity: $a + (b + c) = (a + b) + c$

   1.2 Commutativity: $a + b = b + a$

   1.3 Additive identity: $\exists 0 \in F, \forall a \in F: a + 0 = 0 + a = a$

   1.4 Additive inverses: $\forall a \in F, \exists {-a} \in F$, called additive inverse of $a$, such that $a + (-a) = 0$

2. $(F \backslash \{0\} , ·)$ is a commutative group, where the neutral element is denoted by $1$.

   2.1 Associativity: $a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c$

   2.2 Commutativity: $a ⋅ b = b ⋅ a$

   2.3 Multiplicative identity: $\exists 1 \in F, \forall a \in F: a ⋅ 1 = 1 ⋅ a = a$

   2.4 Multiplicative inverses: $\forall a \neq 0 \in F, \exists a^{-1} \in F$, called multiplicative inverse of $a$, such that $a ⋅ (a^{-1}) = 1$

3. Distributivity

   $\forall a, b, c \in F: a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)$

This may be summarized by saying: a field has two operations, called addition and multiplication; it is an abelian group under addition with $0$ as the additive identity; the nonzero elements are an abelian group under multiplication with $1$ as the multiplicative identity; and multiplication distributes over addition.

Even more summarized: a field is a commutative ring where $0 \neq 1$, and all nonzero elements are invertible under multiplication.

If $(F, +, ·)$ is a field and $F' ⊂ F$ is a subset of $F$ such that the restriction of addition and multiplication to $F'$ define a field with addition $+ : F' × F' → F'$ and multiplication $· : F' × F' → F'$ on $F'$, then $(F', +, ·)$ is called a subfield of $(F, +, ·)$ and $(F, +, ·)$ is called an extension field of $(F', +, ·)$.

We call $(F, +)$ the additive group of the field. We use the notation $F^{*} := F \backslash \{0\}$ for the set of all elements excluding the neutral element of addition, called $(F^{*}, ·)$ the multiplicative group of the field.

The characteristic of a field $F$, represented as $char(F)$, is the smallest natural number $n ≥ 1$ for which the $n$-fold sum of the multiplicative neutral element $1$ equals zero, i.e. for which $\sum_{i=1}^{n}1 = 0$. If such an $n > 0$ exists, the field is said to have a finite characteristic. If, on the other hand, every finite sum of $1$ is such that it is not equal to zero, then the field is defined to have characteristic 0.

Examples of fields:

• Set of rational numbers $Q$ together with the usual definition of addition and multiplication. Since there is no natural number $n ∈ N$ such that $\sum_{i=1}^{n}1 = 0$ in the set of rational numbers, the characteristic of the field $Q$ is given by $char(Q) = 0$.

• The remainder class set $Z_p$if $p$ is a prime numer. Moreover, since $\sum_{i=1}^{p}1 = 0$ in $Z_p$, we know that those fields have the finite characteristic $p$. To distinguish prime fields from arbitrary modular arithmetic rings, we write $(F_p, +, ·)$ for the ring of modular $p$ arithmetics and call it the prime field of characteristic $p$.