= Prime representation via an ideal class
{title2=$q_I(x,y)=N(xe_1+ye_2)/N(I)$}
Let $I$ be an <integral ideal> of an imaginary quadratic <number field> and $(e_1,e_2)$ an <integral basis> of $I$. Its normalized norm is a positive definite <binary quadratic form>. An unramified rational prime $p$ is represented by this form exactly when there is an integral <prime ideal> $P$ of norm $p$ with $[P]=[I]^{-1}$. Indeed, a representation gives an element $\alpha\in I$ with $N\alpha=pN(I)$, so $P=(\alpha)I^{-1}$ is integral of norm $p$. Conversely, $IP=(\alpha)$ with $\alpha\in I$ produces a representation by expressing $\alpha$ in the basis. <Artin reciprocity> then detects this <ideal class> by Frobenius in the <Hilbert class field>.
Back to article page
