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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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Rational numbers are a mathematical field with specific axioms, allowing for the operations of addition and multiplication
Each rational number has an additive inverse and, with the exception of zero, a multiplicative inverse
The field of rational numbers has no non-trivial field automorphisms, making it unique in its structure
Rational numbers form an ordered field, allowing for comparison between any two numbers
Rational numbers are infinitely dense, with an infinite number of numbers between any two given rational numbers
Any countable, densely ordered set without endpoints can be mapped onto the rational numbers while preserving the order
The set of rational numbers can be put into a one-to-one correspondence with the set of natural numbers
Rational numbers can be represented as points on a square lattice, but this representation includes duplicates
These trees are structured to produce each rational number exactly once, providing a unique representation of rational numbers
Rational numbers form a dense subset of real numbers, meaning there are rational numbers in any interval of real numbers
Rational numbers possess an order topology, a subspace topology, and a metric topology, making them a topological field
Rational numbers are not locally compact and are not complete with respect to the absolute difference metric, with the real numbers serving as their completion
Rational numbers can be extended beyond the usual absolute value metric by introducing alternative metrics, such as the p-adic metrics
The completion of the p-adic metric on rational numbers yields the p-adic number field
This theorem shows that any non-trivial absolute value on rational numbers is equivalent to either the usual real absolute value or a p-adic absolute value, highlighting the significance of rational numbers in number theory
00
In the field of rational numbers, every number except ______ has a multiplicative inverse.
zero
01
The field of rational numbers is characterized by having no non-trivial ______ automorphisms.
field
02
As the smallest field with characteristic ______, ℚ contains no proper subfields.
zero
