= 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.
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