Euclidean field

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). 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)