Field/Focus on R/Introduction/Section

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A set is called a field if there are two binary operations (called addition and multiplication)

and two different elements , which fulfill the following properties.

  1. Axioms for the addition:
    1. Law of associativity: holds for all .
    2. Law of commutativity: holds for all .
    3. is the neutral element of the addition, i.e. holds for all .
    4. Existence of the negative: For every , there exists an element with .
  2. Axioms of the multiplication:
    1. Law of associativity: holds for all .
    2. Law of commutativity: holds for all .
    3. is the neutral element for the multiplication, i.e. holds for all .
    4. Existence of the inverse: For every with , there exists an element such that .
  3. Law of distributivity: holds for all .

It is known from school that all these axioms hold for the real numbers (and for the rational numbers) together with the natural operations.

In a field, we use the convention that multiplication connects stronger than addition. Hence, we write instead of . To further simplify the notation, the product sign is usually omitted. The special elements and in a field are called (the) zero and (the) one. By definition, they have to be different.

For us, the most important examples for a field are the field of rational numbers, the field of real numbers and the field of complex numbers (to be introduced later).

Suppose is a field. Then for every element the element fulfilling is uniquely determined. For the element fulfilling is also uniquely determined.

Let be given and suppose that and are elements fulfilling . Then

which means altogether . For the second part see exercise.

We are trying to find a structure of a field on the set . If is supposed to be the neutral element of the addition and the neutral element of the multiplication, then everything is already determined: The equation must hold, since has an inverse element with respect to the addition, and since holds, due to fact. Hence the operation tables look like


With some tedious computations, one can check that this is indeed a field.

Let be a field,

and let denote elements from . Then the following statements hold.
  1. (annulation rule).
  2. (rules for sign).

  3. From one can deduce or .
  4. (general law of distributivity).
  1. We have . Subtracting (meaning addition with the negative of ) on both sides gives the claim.
  2. See exercise.
  3. See exercise.
  4. See exercise.
  5. We prove this by contradiction, so we assume that and are both not . Then there exist inverse elements and and hence . On the other hand, we have by the premise and so the annulation rule gives

    hence , which contradicts the field properties.

  6. This follows with a double induction, see exercise.