Exercises on rational numbers

Exercise 1. Prove that Q is a field

  1. 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.

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