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(ζn), ζn = exp (2πi / n)
is a number field obtained from Q by adjoining a primitive nth root of unity ζn. 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).

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