Fields are sets where one can do arithmetic "normally", in the sense that one has addition, subtraction, multiplication, and division, except for division by zero. Moreover, these operations satisfy the usual axioms such as the commutative and associative laws for addition and multiplication.

Familiar examples of fields are the rational numbers $\Q$, the real numbers $\R$, and the complex numbers $\C$. For number theory, historically concerned with questions over $\Q$, it was useful to be able to factor particular polynomials that are irreducible over $\Q$. This naturally led to the idea of a global number field, the smallest field contained in $\C$ containing $\Q$ and the root of one polynomial. A global number field is always a finite extension of $\Q$.

Local number fields, which are finite extensions of $p$-adic numbers, are useful in understanding global number fields. Along with $\R$ and $\C$, they are the completions of global number fields with respect to metrics arising from non-trivial absolute values.

The LMFDB contains databases of both local and global number fields. For a given degree and prime $p$, there are only finitely many local number fields, making them ideal for tabulation. Global number fields come from various complete sets based on the degree and either a bound on the absolute discriminant, the set of ramifying primes, or the root discriminant of the Galois closure.

The term *number field* distinguishes these fields from
global and local function fields. These fields share many properties
with number fields, many proven and some conjectural. Global function
fields arise as the field of rational functions on a curve over a
finite field, and local function fields are their completions.