Introduction

By definition, a number field is a finite extension of the rationals $\mathbf{Q}$.

In Hecke, a number field $L$ is recursively defined as being either the field of rational numbers $\mathbf{Q}$ or a finite extension of a number field $K$.

We support two presentations for an extension $L/K$:

• as a quotient $L=K[x]/(f)$, where $f\in K[x]$ is an irreducible polynomial (i.e. a simple extension), or

• as a quotient $L=K[x_1,\dots ,x_n]/(f_1(x_1),\dots ,f_n(x_n))$, where $f_1,\dots ,f_n\in K[x_1,\dots ,x_n]$ are appropriate choices of univariate polynomials making $L$ a field (i.e. a non-simple extension).

In both cases we refer to $K$ as the base field of the number field $L$.

Info

By the Primitive Element Theorem, any finite separable extension $L/K$ can be written as a simple extension $L=K(\alpha )$ for some $\alpha \in L$. The distinction between simple and non-simple extensions is therefore purely computational: it dictates how data is stored and how functions are implemented.

Absolute and Relative Extensions

A useful dichotomy comes from the origin of the base field in the definition of a number field $L/K$. We call $L$ an absolute number field if the base field is equal to the rational numbers $\mathbf{Q}$. We call $L$ a relative number field if the base field is strictly larger than $\mathbf{Q}$.

There are (at least) four concrete types that can be used in the implementation of a number field:

• AbsSimpleNumField for absolute simple number fields $\mathbf{Q}(\alpha )/\mathbf{Q}$,

• AbsNonSimpleNumField for absolute non-simple number fields $\mathbf{Q}({\alpha }_1,...,{\alpha }_n)/\mathbf{Q}$,

• RelSimpleNumField for simple relative extensions $K(\alpha )/K$,

• RelNonSimpleNumField for non-simple relative extensions $K({\alpha }_1,...,{\alpha }_n)/K$.

Example

We can construct an absolute simple quadratic field $K/\mathbb{Q}$ and a non-simple relative extension $L/K$.

julia> K, a = quadratic_field(5)
(Real quadratic field defined by x^2 - 5, sqrt(5))

julia> Kx, x = K["x"]
(Univariate polynomial ring in x over K, x)

julia> L, b = number_field([x^2-2, x^2-3], "b")
(Relative non-simple number field of degree 4 over K, RelNonSimpleNumFieldElem{AbsSimpleNumFieldElem}[b1, b2])

julia> typeof(K)
AbsSimpleNumField

julia> typeof(L)
RelNonSimpleNumField{AbsSimpleNumFieldElem}

Both the absolute and relative simple number field types are concrete subtypes of the abstract type SimpleNumField{T} parametrized by the element type T of the associated base field. Both absolute and relative non-simple number field types are subtypes of the abstract type NonSimpleNumField{T} parametrized similarly. These types are themselves subtypes of the abstract parametrized type NumField{T}.

Thus a (simplified) graph of the type tree for number fields is:

NumField{QQFieldElem}                   NumField{T}
├── NonSimpleNumField{QQFieldElem}      ├── NonSimpleNumField{T}
│   └── AbsNonSimpleNumField            │   └── RelNonSimpleNumField
└── SimpleNumField{QQFieldElem}         └── SimpleNumField{T}
    └── AbsSimpleNumField                   └── RelSimpleNumField

Elements of fields implemented by one of these concrete types have a similar type but with an Elem suffix (e.g. an element of an absolute simple number field of type AbsSimpleNumField has type AbsSimpleNumFieldElem).