Difference between revisions of "Euclidean field"
From Encyclopedia of Mathematics
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| − | An ordered field in which every positive element is a square. For example, the field | + | {{TEX|done}} |
| + | 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 | + | 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
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