Normal Quadratics in Ore Extensions, Quantum Planes, and Quantized Weyl Algebras

Let R be a Noetherian domain and let ({\sigma}, {\delta}) be a quasi-derivation of R such that {\sigma} is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F = t^2 - v is normal in the Ore extension R[t; {\sigma}, {\delta}] where v {\epsilon} R. We show F is prime in R[t; {\sigma}, {\delta}] if and only if F is irreducible in Q[t; {\sigma}, {\delta}] if and only if there does not exist w {\epsilon} Q such that v = {\sigma} (w)w - {\delta} (w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.