Coprime discriminants imply linear disjointness

Two finite Galois extensions of the rational number field with coprime field discriminants have trivial intersection. Any ramified prime in the intersection would ramify in both fields. The intersection is therefore everywhere unramified at finite primes and has field discriminant of absolute value one. The Minkowski lower bound for a number-field discriminant excludes this in degree greater than one. The Galois correspondence then gives the product degree for their compositum. The normality hypothesis is used in this last intersection-to-degree argument.