An elementary proof of the theorem on the imaginary quadratic fields with class number 1

Let D be a square-free integer other than 1. Let K be the quadratic field ℚ(√(D)). Let δ∈{1,2} with δ=2 if D≡ 1 4. To each prime ideal 𝒫 in K that splits in K/ℚ we associate a binary quadratic form f_𝒫 and show that when K is imaginary then 𝒫 is principal if and only if f_𝒫 represents δ^2, and when K is real then 𝒫 is principal if and only if f_𝒫 represents ±δ^2. As an application of this result we obtain an elementary proof of the well-known theorem on the imaginary quadratic fields with class number 1. The proof reveals some new information regarding necessary conditions for an imaginary quadratic field to have class number 1 when D≡ 1 4.