Rational numbers (ℚ) are a foundational element in mathematics, forming a field with unique structural characteristics. They are countable, densely ordered, and serve as a model for countable ordered sets. ℚ is also extendable to p-adic numbers, highlighting its importance in number theory. The text explores ℚ's algebraic structure, order properties, countability, topological aspects, and its extension to p-adic numbers.
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1
In the field of rational numbers, every number except ______ has a multiplicative inverse.
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2
The field of rational numbers is characterized by having no non-trivial ______ automorphisms.
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3
As the smallest field with characteristic ______, ℚ contains no proper subfields.
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4
Ordered field property of ℚ
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5
Countable and densely ordered sets mapping to ℚ
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6
Existence of rationals between fractions a/b and c/d
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7
A ______ lattice is used to demonstrate how each rational number corresponds to a unique point, represented as a fraction.
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8
The Calkin–Wilf tree and the Stern–Brocot tree avoid duplicates by ensuring each ______ number appears only once.
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9
In the realm of real numbers, rational numbers form a ______ set, indicating that irrational numbers are uncountably infinite.
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10
The set of ______ numbers is considered 'larger' in measure compared to the set of ______ numbers.
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11
Density of rational numbers in real numbers
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12
Rational numbers as continued fractions
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13
Completeness of rational numbers
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14
The set of ______ numbers can be expanded with alternative metrics, like the ______ metrics for each ______ number.
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15
Ostrowski's theorem states that any non-trivial absolute value on the ______ numbers is equivalent to the usual real absolute value or a ______ absolute value.
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16
The theorem by ______ highlights the intricate structure of ______ numbers and their importance in ______ theory.
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17
The ______ numbers serve as a basis for constructing various complete ______ fields, as demonstrated by the ______ theorem.
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