Euclidean field
In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some y in K.
Properties
- Every Euclidean field is an ordered Pythagorean field, but the converse is not true.[1]
- If E/F is a finite extension, and E is Euclidean, then so is F. This "going-down theorem" is a consequence of the Diller–Dress theorem.[2]
Examples
- The real numbers R with the usual operations and ordering form a Euclidean field.
- The field of real algebraic numbers is a Euclidean field.
- The real constructible numbers, those (signed) lengths which can be constructed from a rational segment by ruler and compass constructions, form a Euclidean field.[3]
- The field of hyperreal numbers is a Euclidean field.
Counterexamples
- The rational numbers Q with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in Q since the square root of 2 is irrational.[4] By the going-down result above, no algebraic number field can be Euclidean.[2]
- The complex numbers C do not form a Euclidean field since they cannot be given the structure of an ordered field.
Euclidean closure
The Euclidean closure of an ordered field K is an extension of K in the quadratic closure of K which is maximal with respect to being an ordered field with an order extending that of K.[5]
gollark: It works in hexchat and irssi (you may need to somehow get it to not use SSL, since it's up but not working properly?).
gollark: The main channel is "#a".
gollark: It lacks:- any security- any reliability also- "NickServ"but oh well!
gollark: It's a network because there are two servers.
gollark: I got bored and decided to make (technically) an IRC network with a friend! So you can connect to irc.osmarks.tk port 6667 (I couldn't make SSL work properly due to some kind of protocol version mismatch).
References
- Martin (1998) p. 89
- Lam (2005) p.270
- Martin (1998) pp. 35–36
- Martin (1998) p. 35
- Efrat (2006) p. 177
- Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs. 124. Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002.
- Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
- Martin, George E. (1998). Geometric Constructions. Undergraduate Texts in Mathematics. Springer-Verlag. ISBN 0-387-98276-0. Zbl 0890.51015.
External links
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