Note that there is a ``largest ideal'' with this condition because if proposition 1 is true for $\mathcal{C}_1,\mathcal{C}_2$ then it is also true for $\mathcal{C}_1+\mathcal{C}_2$ .

Note: It can be proved that there is a unique ray class field with a given conductor. In words, the ray class field is the biggest abelian extension of $K$ with a given conductor (although the conductor of $K_{\mathcal{I}}$ does not necessarily equal $\mathcal{I}$ !, see example $2$ ).

Remark: Let $\mathfrak{p}$ be a prime of $K$ unramified in $L$ , and let $\mathfrak{P}$ be a prime above $\mathfrak{p}$ . Then $(\mathfrak{p},L/K)=1$ if and only if the extension of residue fields is of degree 1 $$[\mathcal{O}_L/\mathfrak{P}\colon \mathcal{O}_K/\mathfrak{p}]=1$$ if and only if $\mathfrak{p}$ splits completely in $L$ . Thus we obtain a characterization of the ray class field of conductor $\mathcal{C}$ as the abelian extension of $K$ such that a prime of $K$ splits completely if and only if it is of the form $$(\alpha),\quad \alpha \in K^{\ast},\ \alpha\equiv1\ \operatorname{mod}\ \mathcal{C}$$

Examples:

1. The ray class field of $\Rats$ of conductor $N\Ints$ is the $N^{th}$ -cyclotomic extension of $\Rats$ . More concretely, let $\zeta_N$ be a primitive $N^{th}$ root of unity. Then $$\Rats_{N\Ints}=\Rats(\zeta_N)$$
2. $$\Rats(i)_{(2)}=\Rats(i)$$ so the conductor of $\Rats(i)_{(2)}/\Rats$ is $(1)$ .
3. $K_{(1)}$ , the ray class field of conductor $(1)$ , is the maximal abelian extension of $K$ which is unramified everywhere. It is, in fact, the Hilbert class field of $K$ .

Bibliography

1

Artin/Tate, Class Field Theory. W.A.Benjamin Inc., New York.