The count of F(s,t) that are norms from such L has a specific order of magnitude derived via the fundamental lemma of sieve theory and geometry of numbers.
Odoni, The Farey density of norm subgroups of global fields (I).Mathematika 20(2): (1973) 155-169
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Solubility for families of norm equations coming from abelian number fields
The count of F(s,t) that are norms from such L has a specific order of magnitude derived via the fundamental lemma of sieve theory and geometry of numbers.