On a Property of the Adele Ring of the Rationals

In order to prove the theorem we intend to apply Lemma 1.2 to the family {R} U {Qp : p e P} of rigid fields. So it suffices to see that the hypothesis (1) of the lemma is satisfied.

For an arbitrary prime p chose a positive integer a with (-) = -1 and a negative integer 6 with (^) = 1. Then a witnesses that there exist no ring monomorphism K —> Qp, while b witnesses that there exist no ring monomorphism Qp —> R.