Algebraic number field

An algebraic number field or a number field for short (old rationality domain) is a finite extension of the field of rational numbers in mathematics . The investigation of algebraic number fields is a central subject of algebraic number theory , a branch of number theory . ${\displaystyle \mathbb {Q} }$

The whole rings of algebraic number fields, which are analogues of the ring of whole numbers in the field , play an important role . ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} }$

Definition and simple properties

An algebraic number field is defined as a finite field extension of the field of rational numbers. This means that the vector space has a finite dimension . This dimension is called the degree of the number field. ${\displaystyle K}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle K}$ ${\displaystyle \mathbb {Q} }$

As finite extensions, number fields are always algebraic extensions of ; that is, every element of a number field is the root of a polynomial with rational coefficients and is therefore an algebraic number . However, not every algebraic extension of is the opposite of a number field: For example, the field of all algebraic numbers is an algebraic, but not a finite extension of , i.e. not an algebraic number field. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {A} }$ ${\displaystyle \mathbb {Q} }$

After the primitive element theorem number fields are simple field extensions of , that can be in the form as adjunction an algebraic number to represent. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} (\xi )}$ ${\displaystyle \xi }$ ${\displaystyle \mathbb {Q} }$

Wholeness

An element of a number field is called whole if it is the zero of a normalized polynomial (leading coefficient 1) with coefficient off . That is, satisfies an equation of shape ${\displaystyle x}$ ${\displaystyle K}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle x}$

${\displaystyle x^{m}+c_{m-1}x^{m-1}+\dotsb +c_{1}x+c_{0}=0}$

with whole numbers . Such numbers are also called whole algebraic numbers . ${\displaystyle c_{0},\dotsc ,c_{m-1}\in \mathbb {Z} }$

The whole numbers form a subring of , called the whole ring of and is usually referred to as , or . ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle {\mathcal {O}}_{K}}$ ${\displaystyle O_{K}}$ ${\displaystyle \mathbb {Z} _{K}}$

Examples

• As a trivial example, there is itself a number field (of degree 1). As expected , d. that is, the whole rational numbers are the "normal" whole numbers. ${\displaystyle \mathbb {Q} }$ ${\displaystyle {\mathcal {O}}_{\mathbb {Q} }=\mathbb {Z} }$
• The field of complex numbers with rational real and imaginary parts is a number field of degree 2. The associated wholeness ring is the ring of (whole) Gaussian numbers . ${\displaystyle \mathbb {Q} (i)=\{a+bi\in \mathbb {C} :a,b\in \mathbb {Q} \}}$ ${\displaystyle \mathbb {Z} [i]=\{a+bi\in \mathbb {C} :a,b\in \mathbb {Z} \}}$
• More generally, the square number fields with square-free form exactly the number fields of degree 2. For the wholeness rings this results ${\displaystyle \mathbb {Q} \left({\sqrt {d}}\right)}$ ${\displaystyle d\in \mathbb {Z} \setminus \{1\}}$

${\displaystyle \mathbb {Z} \left[{\sqrt {d}}\right]}$if congruent 2 or 3 is mod 4, ${\displaystyle d}$

${\displaystyle \mathbb {Z} \left[{\tfrac {1+{\sqrt {d}}}{2}}\right]}$if congruent 1 is mod 4. ${\displaystyle d}$

• The fields of circular division with a primitive -th root of unit are number fields of degree  with Euler's φ-function . The wholeness ring is . ${\displaystyle \mathbb {Q} (\zeta _{n})}$ ${\displaystyle n}$ ${\displaystyle \zeta _{n}}$ ${\displaystyle \varphi (n)}$ ${\displaystyle \mathbb {Z} [\zeta _{n}]}$

Bases

Since a number field of degree is  a -dimensional vector space, every basis of consists of exactly elements. If there is such a basis, then each element can be written in the form ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle K}$ ${\displaystyle n}$ ${\displaystyle \{x_{1},\dotsc ,x_{n}\}}$ ${\displaystyle x\in K}$

${\displaystyle x=a_{1}x_{1}+\dotsb +a_{n}x_{n}}$

with clearly defined coefficients , which, however, depend on the choice of the base. Hold , then has the special basis , where the degree of is equal to the degree of the minimal polynomial of the algebraic number . ${\displaystyle a_{j}\in \mathbb {Q} }$ ${\displaystyle K=\mathbb {Q} (\xi )}$ ${\displaystyle K}$ ${\displaystyle \left\{1,\xi ,\xi ^{2},\dotsc ,\xi ^{n-1}\right\}}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle \xi }$

A base of is integral basis when each whole element in the form of can be written. For example, there is a basis of , but not a wholeness basis, because not all elements of the wholeness ring can be written as integer linear combinations of 1 and . In contrast, there is a holistic basis of . ${\displaystyle K}$ ${\displaystyle x\in {\mathcal {O}}_{K}}$ ${\displaystyle x=a_{1}x_{1}+\dotsb +a_{n}x_{n}}$ ${\displaystyle a_{1},\dotsc ,a_{n}\in \mathbb {Z} }$ ${\displaystyle \left\{1,{\sqrt {5}}\right\}}$ ${\displaystyle \mathbb {Q} \left({\sqrt {5}}\right)}$ ${\displaystyle \mathbb {Z} \left[{\tfrac {1+{\sqrt {5}}}{2}}\right]}$ ${\displaystyle {\sqrt {5}}}$ ${\displaystyle \left\{1,{\tfrac {1+{\sqrt {5}}}{2}}\right\}}$ ${\displaystyle \mathbb {Q} \left({\sqrt {5}}\right)}$

Another base-dependent representation of elements of a number field is the matrix representation . Is to select fixed, then by multiplication with a linear transformation , given. This linear operator can be relative to a fixed base by a square matrix representing . The determinant and the trace of the mapping (i.e. the representing matrix), which are independent of the choice of the base, are called norm and trace of and are important tools for calculations and proofs in algebraic number fields. ${\displaystyle K}$ ${\displaystyle x\in K}$ ${\displaystyle x}$ ${\displaystyle A_{x}\colon K\to K}$ ${\displaystyle A_{x}(z)=x\cdot z}$ ${\displaystyle x}$

Generalization and classification

The algebraic number field together with the functional bodies of the characteristics of the class of the global body , which together with the local bodies , which include for instance the body of p -adischen numbers represent are the most important research objects of algebraic number theory. ${\displaystyle \mathbb {F} _{p}(T)}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Q} _{p}}$

See also

• Ideal class group
• Dirichletscher unit set
• Sorrow extension
• Quadratic law of reciprocity , Artin's law of reciprocity
• Class field theory
• Brewers group
• Iwasawa theory
• Dedekind's zeta function

literature

• Stefan Müller-Stach , Jens Piontkowski: Elementary and algebraic number theory. A modern approach to classic topics. Vieweg, Wiesbaden 2006, ISBN 3-8348-0211-5 ( Vieweg studies ).
• Jürgen Neukirch : Algebraic number theory . Springer-Verlag, Berlin 2006. ISBN 3-540-37547-3 .