Abstract
We investigate the connections between the values assumed by binary quadratic forms over a field F (of characteristic not 2) and certain 2-groups arising as Galois groups over F. The groups in question will always be quotients of the so-called W-group of F. This group is the Galois group of the compositum over F of all quadratic extensions, cyclic extensions of order 4, and dihedral extensions of order 8. We show how the W-group and its quotients determine the values assumed by any binary form. The main result is to apply these ideas to give a simple characterization via Galois groups of fields with level ≤4.
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