3 - Rational numbers

Summary

This chapter treats the different kinds of structures on the field ℚ of rational numbers (algebra, order and topology) and various combinations of them in the same way as it was done in Chapter 1 for the field ℝ of real numbers.

We sometimes profit from the fact that ℚ is embedded in ℝ so that we can use results from Chapter 1. In doing so, we take the field ℝ for granted. Constructions of ℝ from ℚ are presented in Chapter 2 (via an ultrapower of ℚ) and in Chapter 4 (by completion of ℚ).

The additive group of the rational numbers

Under the usual addition, the rational numbers form a group (ℚ, +), or briefly ℚ⁺. Being a subgroup of ℝ⁺, this group has already been studied to some extent in Section 1, together with the factor group ℚ⁺/ℤ. We continue the investigation of these groups, we characterize them in the class of all groups, and we study their endomorphism rings.

Definition A group G is called locally cyclic, if the subgroup 〈a₁, a₂, …, a_n〉 generated by finitely many elements a₁, a₂, …, a_n of G is always cyclic, that is, this subgroup may be generated in fact by a single element of G. By induction, this is equivalent to the property that the subgroup generated by any two elements is cyclic. In particular, every locally cyclic group is abelian.

For 0 ≠ b ∈ ℚ the subgroup 〈b〉 of ℚ⁺generated by b is isomorphic to 〈1〉 = ℤ⁺, as there is an automorphism of ℚ⁺mapping b to 1.