Exercises on rational numbers
Exercise 1. Prove that Q is a field
We prove first that ℚ is a commutative ring with unit: (b,a) ⋅ (a,b) = (ba, ab) = (ab, ba) = (a,b) ⋅ (b,a). This relations holds true since ℕ is commutative. (1,1) is the unit since (a,b) ⋅ (1,1) = (a,b), by the definition of the multiplication operation in ℚ. What is left to prove is the existence of the multiplicative inverse for each element in ℚ:
(a,b) ⋅ (b,a) = (1,1), ∀(a,b) such that a ≠ 0, b ≠ 0
by construction ℚ := (ℤ x ℤ \ {0})/ρ, so a ≠ 0, b ≠ 0 is always satisfied and every element has multiplicative inverse.
Note. The set ℚ is not a group with respect to multiplication. The rational number 0 ∈ ℚ has no multiplicative inverse 0a = 1.