On indefinite and potentially universal quadratic forms over number fields

A number field k k admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of k k is bigger than one. In this case, there are only finitely many classes of such binary integral quadratic forms over k k . A number field k k admits a ternary integral quadratic form which represents all integers locally but not globally if and only if the class number of k k is even. In this case, there are infinitely many classes of such ternary integral quadratic forms over k k . An integral quadratic form over a number field k k with more than one variables represents all integers of k k over the ring of integers of a finite extension of k k if and only if this quadratic form represents 1 1 over the ring of integers of a finite extension of k k .