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number field (Definition)
Definition 1   A field which is a finite extension of $ \mathbb{Q}$, the rational numbers, is called a number field (sometimes called algebraic number field). If the degree of the extension $ K/\mathbb{Q}$ is $ n$ then we say that $ K$ is a number field of degree $ n$ (over $ \mathbb{Q}$).
Example 1   The field of rational numbers $ \mathbb{Q}$ is a number field.
Example 2   Let $ K=\mathbb{Q}(\sqrt{d})$, where $ d\neq 1$ is a square-free non-zero integer and $ \sqrt{d}$ stands for any of the roots of $ x^2-d=0$ (note that if $ \sqrt{d}\in K$ then $ -\sqrt{d}\in K$ as well). Then $ K$ is a number field and $ [K:\mathbb{Q}]=2$. We can explictly describe all elements of $ K$ as follows:
$\displaystyle K=\{ t+s\sqrt{d} : t,s \in \mathbb{Q}\}.$
Definition 2   A number field $ K$ such that the degree of the extension $ K/\mathbb{Q}$ is $ 2$ is called a quadratic number field.

In fact, if $ K$ is a quadratic number field, then it is easy to show that $ K$ is one of the fields described in Example $ 2$.

Example 3   Let $ K_n=\mathbb{Q}(\zeta_n)$ be a cyclotomic extension of $ \mathbb{Q}$, where $ \zeta_n$ is a primitive $ n$th root of unity. Then $ K$ is a number field and
$\displaystyle [K:\mathbb{Q}]=\varphi(n)$
where $ \varphi(n)$ is the Euler phi function. In particular, $ \varphi(3)=2$, therefore $ K_3$ is a quadratic number field (in fact $ K_3=\mathbb{Q}(\sqrt{-3})$). We can explicitly describe all elements of $ K$ as follows:
$\displaystyle K_n=\{ q_0+q_1\zeta_n+q_2\zeta_n^2+\ldots+q_{n-1}\zeta_n^{n-1} : q_i\in \mathbb{Q}\}.$
In fact, one can do better. Every element of $ K_n$ can be uniquely expressed as a rational combination of the $ \varphi(n)$ elements $ \{\zeta_n^a : \gcd(a,n)=1,\ 1\leq a < n\}$.
Example 4   Let $ K$ be a number field. Then any subfield $ L$ with $ \mathbb{Q}\subseteq L \subseteq K$ is also a number field. For example, let $ p$ be a prime number and let $ F=\mathbb{Q}(\zeta_p)$ be a cyclotomic extension of $ \mathbb{Q}$, where $ \zeta_p$ is a primitive $ p$th root of unity. Let $ F^+$ be the maximal real subfield of $ F$. $ F^{+}$ is a number field and it can be shown that:
$\displaystyle F^+=\mathbb{Q}(\zeta_p+\zeta_p^{-1}).$



"number field" is owned by alozano. [ full author list (2) | owner history (1) ]
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See Also: algebraic number theory, examples of prime ideal decomposition in number fields, examples of fields, abelian extensions of quadratic imaginary number fields, number theory, residue degree, regulator, discriminant ideal, class number, existence of Hilbert class field, multiplicatively congruent, class number formula, examples of totally real fields, prime ideal decomposition in quadratic extensions of $\mathbb{Q}$, valuation, prime, unramified, ideal class, ray class field, units of quadratic fields, calculating the splitting of primes, fundamental units, Eisenstein integers, Kronecker-Weber theorem, ramification index, examples of ring of integers of a number field, ideal classes form an abelian group, table of some fundamental units

Other names:  algebraic number field
Also defines:  quadratic number field, quadratic field

Attachments:
abelian number field (Definition) by alozano
quadratic fields that are not isomorphic (Theorem) by Wkbj79
imaginary quadratic field (Definition) by pahio
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Cross-references: maximal real subfield, prime number, subfield, combination, rational, Euler phi function, root of unity, primitive, cyclotomic extension, roots, integer, square-free, extension, degree, rational numbers, finite extension, field
There are 118 references to this entry.

This is version 13 of number field, born on 2001-12-21, modified 2006-10-15.
Object id is 1128, canonical name is NumberField.
Accessed 11609 times total.

Classification:
AMS MSC: 11-00 (Number theory :: General reference works )
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