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Difference between revisions of "Euclidean field"

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(MSC 12J15 11R04)
(→‎Comments: Euclidean vs norm-Euclidean)
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====Comments====
 
====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|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.
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There is a second meaning in which the phrase Euclidean field is used (especially for quadratic number fields). An [[algebraic 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]], and ''norm-Euclidean'' if it Euclidean with respective to the field norm from $K$ to $\mathbf{Q}$. The norm-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: the field with $m = 14$ is Euclidean but not norm-Euclidean and it is conjectured that there are infinitely many Euclidean quadratic fields with $m > 0$.  It is known that there are no further Euclidean quadratic fields with $m < 0$, cf. [[#References|[b1]]], Chapt. 14.
  
 
====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 22:16, 24 November 2016

2020 Mathematics Subject Classification: Primary: 12J15 Secondary: 11R04 [MSN][ZBL]

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). An algebraic 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, and norm-Euclidean if it Euclidean with respective to the field norm from $K$ to $\mathbf{Q}$. The norm-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: the field with $m = 14$ is Euclidean but not norm-Euclidean and it is conjectured that there are infinitely many Euclidean quadratic fields with $m > 0$. It is known that there are no further Euclidean quadratic fields with $m < 0$, cf. [b1], Chapt. 14.

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=39808
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article