By definition, mathematically a number field is just a finite extension of the rational $\mathbf{Q}$. In Hecke, a number field $L$ is recursively defined as being the field of rational numbers $\mathbf{Q}$ or a finite extension of a number field $K$. In the second case, the extension can be defined in the one of the following two ways:

  • We have $L = K[x]/(f)$, where $f \in K[x]$ is an irreducible polynomial (simple extension), or
  • We have $L = K[x_1,\dotsc,x_n]/(f_1(x_1),\dotsc,f_n(x_n))$, where $f_1,\dotsc,f_n \in K[x]$ are univariate polynomials (non-simple extension).

In both cases we refer to $K$ as the base field of the number field $L$. Another useful dichotomy comes from the type of the base field. We call $L$ an absolute number field, if the base field is equal to the rational numbers $\mathbf{Q}$.