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form a
Euclidean field. It is the smallest Euclidean field, as every Euclidean field contains it as an ordered subfield. In other words, the constructible numbers form the
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This article is about ordered fields. For algebraic number fields whose ring of integers has a
Euclidean algorithm, see
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with the usual operations and ordering do not form a
Euclidean field. For example, 2 is not a square in
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do not form a
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which is maximal with respect to being an ordered field with an order extending that of
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is a
Euclidean field. The following examples are also real closed fields.
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with the usual operations and ordering form a
Euclidean field.
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for which every non-negative element is a square: that is,
237:{\displaystyle \mathbb {R} \cap \mathbb {\overline {Q}} }
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Ordered field where every nonnegative element is a square
34:. For the class of models in statistical mechanics, see
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144:. This "going-down theorem" is a consequence of the
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501:Introduction to Quadratic Forms over Fields
381:. It is also the smallest subfield of the
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505:Graduate Studies in Mathematics
164:ruler and compass constructions
121:, but the converse is not true.
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471:American Mathematical Society
434:Martin (1998) pp. 35–36
341:{\displaystyle \mathbb {C} }
301:{\displaystyle \mathbb {Q} }
279:{\displaystyle \mathbb {Q} }
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196:{\displaystyle \mathbb {R} }
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538:Martin, George E. (1998).
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140:is Euclidean, then so is
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146:Diller–Dress theorem
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119:Pythagorean field
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16:(Redirected from
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401:References
314:irrational
308:since the
112:Properties
391:extension
230:¯
222:∩
158:The real
134:extension
81:for some
596:Category
499:(2005).
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369:in the
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