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Difference between revisions of "Euclidean field"

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An ordered field in which every positive element is a square. For example, the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363401.png" /> of real numbers is a Euclidean field. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363402.png" /> of rational numbers is not a Euclidean field.
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An ordered field in which every positive element is a square. For example, the field $\mathbf R$ of real numbers is a Euclidean field. The field $\mathbf Q$ of rational numbers is not a Euclidean field.
  
  
  
 
====Comments====
 
====Comments====
There is a second meaning in which the phrase Euclidean field is used (especially for quadratic number fields). A number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363403.png" /> (i.e. a finite field extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363404.png" />) is called Euclidean if its ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363405.png" /> is a [[Euclidean ring|Euclidean ring]]. The Euclidean quadratic fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363407.png" /> a square-free integer, are precisely the fields with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363408.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e0363409.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e03634010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e03634011.png" />, 5, 6, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e03634012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036340/e03634013.png" />, 13, 17, 19, 21, 29, 33, 37, 41, 57, or 73, cf. [[#References|[a1]]], Chapt. VI.
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There is a second meaning in which the phrase Euclidean field is used (especially for quadratic number fields). A number field $K$ (i.e. a finite field extension of $\mathbf Q$) is called Euclidean if its ring of integers $A$ is a [[Euclidean ring|Euclidean ring]]. The Euclidean quadratic fields $\mathbf Q(\sqrt m)$, $m$ a square-free integer, are precisely the fields with $m$ equal to $-1$, $\pm2$, $\pm3$, 5, 6, $\pm7$, $\pm11$, 13, 17, 19, 21, 29, 33, 37, 41, 57, or 73, cf. [[#References|[a1]]], Chapt. VI.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weiss,  "Algebraic number theory" , McGraw-Hill  (1963)</TD></TR></table>

Revision as of 16:03, 17 April 2014

An ordered field in which every positive element is a square. For example, the field $\mathbf R$ of real numbers is a Euclidean field. The field $\mathbf Q$ of rational numbers is not a Euclidean field.


Comments

There is a second meaning in which the phrase Euclidean field is used (especially for quadratic number fields). A number field $K$ (i.e. a finite field extension of $\mathbf Q$) is called Euclidean if its ring of integers $A$ is a Euclidean ring. The Euclidean quadratic fields $\mathbf Q(\sqrt m)$, $m$ a square-free integer, are precisely the fields with $m$ equal to $-1$, $\pm2$, $\pm3$, 5, 6, $\pm7$, $\pm11$, 13, 17, 19, 21, 29, 33, 37, 41, 57, or 73, cf. [a1], Chapt. VI.

References

[a1] E. Weiss, "Algebraic number theory" , McGraw-Hill (1963)
How to Cite This Entry:
Euclidean field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_field&oldid=17914
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article