number field

A number field is a subfield of C having finite degree (dimension as a vector space) over Q.

We know  that every such field has the form Q[α] for some algebraic number α∈C.

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If α is a root of an irreducible polynomial over Q having degree n, then Q[α] = {a0+a1α+···+an−1αn−1:  ai∈Q∀i} and representation in this form is unique; in other words, {1,α,…,αn−1} is a basis for Q[α] as a vector space over Q.

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Cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers.

We have already considered the field Q[ω] where ω = e^(2πi/p), p prime. Recall that n = p−1 in that case. More generally, let ω = e^(2πi/m), m not necessarily prime. The field Q[ω] is called the m^th cyclotomic field.

We will show that the cyclotomic fields, for m even (m>0), are all distinct.

The degree of the m^th cyclotomic field over Q is φ(m), the number of elements in the set

{k: 1≤  k ≤ m, (k,m) = 1}.

Another infinite class of number fields consists of the quadratic fields Q[√m], m∈Z, m not a perfect square. (have degree 2 over Q, having basis {1,√m})

There is infinite class of number fields according to your degree.

We need only consider squarefree m since, for example, Q[√12] = Q[√3]. The Q[√m],  for m squarefree, are all distinct.