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). 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)
[b1]	G.H. Hardy; E.M. Wright. "An Introduction to the Theory of Numbers", Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press (2008) [1938] ISBN 978-0-19-921986-5 Zbl 1159.11001