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A [[Field|field]] consisting of complex (e.g., real) numbers. A set of complex numbers forms a number field if and only if it contains more than one element and with any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679102.png" /> their difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679103.png" /> and quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679104.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679105.png" />). Every number field contains infinitely many elements. The field of rational numbers is contained in every number field.
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A [[Field|field]] consisting of complex (e.g., real) numbers. A set of complex numbers forms a number field if and only if it contains more than one element and with any two elements $\alpha$ and $\beta$ their difference $\alpha-\beta$ and quotient $\alpha/\beta$ ($\beta\neq0$). Every number field contains infinitely many elements. The field of rational numbers is contained in every number field.
Examples of number fields are the fields of rational numbers, real numbers, complex numbers, or Gaussian numbers (cf. [[Gauss number|Gauss number]]). The set of all numbers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679107.png" />, forms a number field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n0679109.png" /> is a fixed complex number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791011.png" /> range over the polynomials with rational coefficients.
 
 
 
  
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Examples of number fields are the fields of rational numbers, real numbers, complex numbers, or Gaussian numbers (cf. [[Gauss number]]). The set of all numbers of the form $H(\alpha)/F(\alpha)$, $F(\alpha)\neq0$, forms a number field, $Q(\alpha)$, where $\alpha$ is a fixed complex number and $H(x)$ and $F(x)$ range over the polynomials with rational coefficients.
  
 
====Comments====
 
====Comments====
An algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791012.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791013.png" /> is an extension of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791014.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791015.png" /> of rational numbers. Alternatively, a number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791016.png" /> is an algebraic number field (of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791017.png" />) if every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791018.png" /> is the root of a polynomial (of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791019.png" />) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067910/n06791020.png" />. A number field that is not algebraic is called transcendental. (Cf. also [[Algebraic number theory|Algebraic number theory]]; [[Extension of a field|Extension of a field]]; [[Transcendental extension|Transcendental extension]].)
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The term "number field" is often used for the specific case of an [[algebraic number field]]: a finite degree [[Extension of a field|extension]] of the field $\mathbf Q$ of rational numbers: every $\alpha\in K$ is the root of a polynomial (of degree at most $n$) over $\mathbf Q$. A number field that is not algebraic is called transcendental.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)  pp. Sects. 4–9</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)  pp. Sects. 4–9 {{ZBL|0115.03601}}</TD></TR>
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<TR><TD valign="top">[b1]</TD> <TD valign="top">  H. Salzmann, T. Grundhöfer, H. Hähl, R. Löwen, "The classical fields. Structural features of the real and rational numbers" Encyclopedia of Mathematics and Its Applications '''112''' Cambridge University Press (2007) ISBN 0-521-86516-6  {{ZBL|1173.00006}}</TD></TR>
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</table>

Latest revision as of 21:36, 10 April 2018

A field consisting of complex (e.g., real) numbers. A set of complex numbers forms a number field if and only if it contains more than one element and with any two elements $\alpha$ and $\beta$ their difference $\alpha-\beta$ and quotient $\alpha/\beta$ ($\beta\neq0$). Every number field contains infinitely many elements. The field of rational numbers is contained in every number field.

Examples of number fields are the fields of rational numbers, real numbers, complex numbers, or Gaussian numbers (cf. Gauss number). The set of all numbers of the form $H(\alpha)/F(\alpha)$, $F(\alpha)\neq0$, forms a number field, $Q(\alpha)$, where $\alpha$ is a fixed complex number and $H(x)$ and $F(x)$ range over the polynomials with rational coefficients.

Comments

The term "number field" is often used for the specific case of an algebraic number field: a finite degree extension of the field $\mathbf Q$ of rational numbers: every $\alpha\in K$ is the root of a polynomial (of degree at most $n$) over $\mathbf Q$. A number field that is not algebraic is called transcendental.

References

[a1] E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9 Zbl 0115.03601
[b1] H. Salzmann, T. Grundhöfer, H. Hähl, R. Löwen, "The classical fields. Structural features of the real and rational numbers" Encyclopedia of Mathematics and Its Applications 112 Cambridge University Press (2007) ISBN 0-521-86516-6 Zbl 1173.00006
How to Cite This Entry:
Number field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Number_field&oldid=14270
This article was adapted from an original article by A.B. Shidlovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article