Difference between a general field and a numerical field

A numerical field is a subclass of a general field, where the elements are numbers.

Field: In algebraic terms, a field refers to any structure that satisfies the properties of a field, not necessarily involving numbers, but possibly more abstract elements like functions, polynomials, or other algebraic entities. It consists of a set of elements and two operations (typically addition and multiplication) that follow specific rules.
Numerical field: A field where the elements are numbers or extensions of numerical fields, often used in number theory. For example, the field of rational numbers \( \mathbb{Q} \), the field of real numbers \( \mathbb{R} \), or the field of complex numbers \( \mathbb{C} \).
In algebraic number theory, the term "numerical field" can also refer to a finite extension of the field of rational numbers \( \mathbb{Q} \), such as \( \mathbb{Q}(\sqrt{2}) \), which is the field obtained by adding \( \sqrt{2} \) to the rationals.

Thus, a numerical field is a specific type of field. In general, every numerical field is a field, but not all fields are numerical fields.

A practical example

Here's a practical example to illustrate the difference between a numerical field and a general field:

Example 1

Consider the field of rational numbers \( \mathbb{Q} \), which consists of all fractions of the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \).

In \( \mathbb{Q} \), you can perform addition, subtraction, multiplication, and division (except division by zero), and these operations satisfy the properties of a field.

Addition: \( \frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6} \)
Multiplication: \( \frac{1}{2} \times \frac{2}{3} = \frac{2}{6} = \frac{1}{3} \)
Division (except by 0): \( \frac{1}{2} \div \frac{2}{3} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \)

This is an example of a numerical field because its elements are numbers, and you can perform operations between them.

Example 2

Now, consider the field of rational functions \( \mathbb{F}(x) \), which consists of all rational functions—ratios of polynomials with coefficients in a field \( \mathbb{F} \).

For example, you can have \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) \neq 0 \).

In this case, the operations are performed on rational functions rather than numbers, but the system still satisfies the properties of a field.

Addition: If \( f(x) = \frac{1}{x+1} \) and \( g(x) = \frac{2}{x+2} \), then \( f(x) + g(x) = \frac{1}{x+1} + \frac{2}{x+2} = \frac{(x+2) + 2(x+1)}{(x+1)(x+2)} = \frac{3x+4}{(x+1)(x+2)} \).
Multiplication: \( f(x) \times g(x) = \frac{1}{x+1} \times \frac{2}{x+2} = \frac{2}{(x+1)(x+2)} \).
Division: \( \frac{f(x)}{g(x)} = \frac{\frac{1}{x+1}}{\frac{2}{x+2}} = \frac{x+2}{2(x+1)} \).

This is an example of a general, non-numerical field because its elements are algebraic expressions (polynomials or ratios of polynomials), not simple numbers.

And so on.