Algebraic Number Field - Examples

Examples

• The smallest and most basic number field is the field Q of rational numbers. Many properties of general number fields, such as unique factorization, are modelled after the properties of Q.

• The Gaussian rationals, denoted Q(i) (read as "Q adjoined i"), form the first nontrivial example of a number field. Its elements are expressions of the form

a+bi

where both a and b are rational numbers and i is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity

i2 = −1.

Explicitly,

(a + bi) + (c + di) = (a + c) + (b + d)i,

(a + bi) (c + di) = (ac − bd) + (ad + bc)i.

Non-zero Gaussian rational numbers are invertible, which can be seen from the identity

It follows that the Gaussian rationals form a number field which is two-dimensional as a vector space over Q.

• More generally, for any square-free integer d, the quadratic field

Q(√d)

is a number field obtained by adjoining the square root of d to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of gaussian rational numbers, d = − 1.

• Cyclotomic field

Q(ζ_n), ζ_n = exp (2πi / n)

is a number field obtained from Q by adjoining a primitive nth root of unity ζ_n. This field contains all complex nth roots of unity and its dimension over Q is equal to φ(n), where φ is the Euler totient function.

• The real numbers, R, and the complex numbers, C, are fields which have infinite dimension as Q-vector spaces, hence, they are not number fields. This follows from the uncountability of R and C as sets, whereas every number field is necessarily countable.

• The set Q2 of ordered pairs of rational numbers, with the entrywise addition and multiplication is a two-dimensional commutative algebra over Q. However, it is not a field, since it has zero divisors:

(1, 0) · (0, 1) = (1 · 0, 0 · 1) = (0, 0).