An Artin character and representations of primes by binary quadratic forms II

Abstract

For any squarefree positive m there exists exactly one solvable antipellian equation, which can be used to construct a certain dihedral extension L/Q, cyclic of degree 4 above k=Q(√−m). We calculate the conductor of L/k and the value of the Artin character of L/k on the corresponding congruence ideal classes of order 2 of k. From this, we deduce results for the representations of powers of primes by binary quadratic forms, in the case where the norm of the fundamental unit of Q(√m) is +1.