The word ‘field’ describes abstractly an algebraic structure in which the operations +, -, × and ÷ may be performed in much the same way as they are performed with the rational numbers. When copies of a field F are used to make cartesian products such as F × F or F × F × F such cartesian products are called vector spaces when furnished with a suitable addition of elements and multiplication by a scalar (i.e. a field element).

Concurrent reading: Birkhoff and MacLane, chapter 2, section 1 and chapter 7, sections 1–4.

Fields

If in a group (G, ·) xy = yx for any two elements x, y ∈ G, then the group G is said to be commutative, or abelian.

1. 1 The rational numbers ℚ, the real numbers ℝ and the complex numbers ℂ each form an abelian group under addition, and, with the deletion of zero, an abelian group under multiplication. These two properties, together with the distributive laws a · (b + c) = a · b + a · c and (a + b) · c = a · c + b · c make (ℚ, +, ·), (ℝ, +, ·) and (ℚ, +, ·) into fields, why do the integers (ℤ, +, ·) not form a field?

2. 2 […]