Let $K$ be a number field and $F$ a polynomial with coefficients in $K$. Class field theory shows that the following properties of $F$ are equivalent:
(1) There is subgroup $H$ of a generalized ideal class group $C_{\mathfrak m}$ (of some appropriately chosen conductor $\mathfrak m$) and a finite set of prime ideals $S$ (including all primes involved in the denominators of $F$) such that $F$ splits completely modulo $\wp$ (for $\wp \not\in S$) if and only if the class of $\wp$ in $C_{\mathfrak m}$ lies in $H$.
(2) The splitting field of $F$ over $K$ is an abelian extension.
In Shimura's example, the field $K$ is $\mathbf Q$, the conductor is $7$, the set $S$ is ${7}$, \{7\}$, and the splitting field is the degree 3 extension of$\mathbf Q$contained in${\mathbf Q}(\zeta_7)$. So in general, to construct$F$of the type considered by Shimura, one must construct abelian extensions of number fields$K$. The theory of complex multiplication is one tool that allows one to do this. 2 added 22 characters in body Let$K$be a number field and$F$a polynomial with coefficients in$K$. Class field theory shows that the following properties of$F$are equivalent: (1) There is subgroup$H$of a generalized ideal class group$C_{\mathfrak m}$(of some appropriately chosen conductor$\mathfrak m$) and a finite set of prime ideals$S$(including all primes involved in the denominators of$F$) such that$F$splits completely modulo$\wp$(for$\wp \not\in S$) if and only if the class of$\wp$in$C_{\mathfrak m}$lies in$H$. (2) The splitting field of$F$over$K$is an abelian extension. In Shimura's example, the field$K$is$\mathbf Q$, the conductor is$7$, the set$S$is${7}$, and the splitting field is the degree 3 extension of$\mathbf Q$contained in${\mathbf Q}(\zeta_7)$. So in general, to construct$F$of the type considered by Shimura, one must construct abelian extensions of number fields$K$. The theory of complex multiplication is one tool that allows one to do this. 1 Let$K$be a number field and$F$a polynomial with coefficients in$K$. Class field theory shows that the following properties of$F$are equivalent: (1) There is subgroup$H$of a generalized ideal class group$C_{\mathfrak m}$(of some appropriately chosen conductor$\mathfrak m$) and a finite set of prime ideals$S$(including all primes involved in the denominators of$F$) such that$F$splits completely modulo$\wp$if and only if the class of$\wp$in$C_{\mathfrak m}$lies in$H$. (2) The splitting field of$F$over$K$is an abelian extension. In Shimura's example, the field$K$is$\mathbf Q$, the conductor is$7$, the set$S$is${7}$, and the splitting field is the degree 3 extension of$\mathbf Q$contained in${\mathbf Q}(\zeta_7)$. So in general, to construct$F$of the type considered by Shimura, one must construct abelian extensions of number fields$K\$. The theory of complex multiplication is one tool that allows one to do this.