Fields

The most useful number systems are those in which we can divide by nonzero elements (we've already touched on the notion of a field). A field is a ring in which the nonzero elements form an abelian group under multiplication. In other words, a field is a nontrivial commutative ring R satisfying the following extra axiom.

For each nonzero element a ∈ R there exists a⁻¹ ∈ R such that a · a⁻¹ = 1.

The rings ℚ, ℝ and ℂ are all fields, but the integers do not form a field.

It was also Dedekind who baptized what we know as fields (with the German word Körper, literally “body”), because it " denotes a system with a certain completeness, fullness and self-containment".

Definition 8.1.1. A fleld is a commutative ring R with identity 1_R ≠ O_R that satisfies this axiom:

Then a field is a set together with two laws of composition:

called addition and multiplication, which satisfy the following axioms:

1. F is an Abelian group under the "addition" operation +. The additive identity is called the zero and is denoted by 0;

2. Multiplication is associative and commutative and makes the set F* = F \ {0} of nonzero elements of F is an Abelian group under the "multiplication" operation '⋅'. The multiplicative identity is called the unity and is denoted by 1;

3. The multiplication operation '⋅' distributes over the addition operation +; that is a ⋅ (b + c) = a ⋅ b + a ⋅ c, for every a,b,c ∈ F.  □

From the definition, we see that a field consists of two groups with respect to two operations, addition and multiplication. The group with respect to addition is called the additive group of the field, and the group with respect to multiplicatoin is called the multiplicative group of the field. Since these two groups must contain an identity element, a field must contain at least two elements. In a field the additive inverse of an element a is denoted by '−a', and the mutliplicative inverse of a is denoted a⁻¹, provided that a ≠ 0.

The properties satisfied by a field F, are more extensive than the four group axioms applied to both binary operations. Recall that ℝ under standard addition is an Abelian group. On the other hand, ℝ under standard multplication is not a group − although it comes very close: the inverse property is not satisfied, and it fails for just one real number, zero. By removing the zero and considering the set ℝ* = ℝ\{0} we have a multiplicative group. Indeed, the first group axiom holds since the product of any two nonzero real numbers is still nonzero. Multiplication of real numbers is associative, 1 plays the role of an identity element, and 1/a gives the inverse of an element a ∈ ℝ*. Thus ℝ* is a group under multiplication.
Similarly, we have the groups ℚ* of all nonzero rational numbers and ℂ* of all nonzero complex numbers, under the operation of ordinary multiplication. If we attempt to form a multiplicative group from the integers ℤ, we have to restrict ourselves to just 1, since these are the only integers that have multiplicative inverses in ℤ.

There are, on the other hand, fields whose elements are not numbers, for example, ℤ_p is a (finite) field for every prime number p.

Subfields

ℤ is a subring of the ring ℚ of rational numbers and ℚ is a subring of the field ℝ of all real numbers. Since ℚ is itself a field, we say that ℚ is a subfietd of ℝ. Similarly, ℝ is a subfield of the field ℂ of complex numbers. Then a subfield of a field K is a subring which is a field. Equivalently, it is a subset F of K satisying the subring Criterion i.e. containing at least two elements, such that