algebraic number
Let an agent handle algebraic numbers by definition, minimal polynomial, degree and related structures in number theory.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent handle algebraic numbers by definition, minimal polynomial, degree and related structures in number theory.
A complex number that is a root of a non-zero polynomial with rational (equivalently integer) coefficients, such as square roots of integers and roots of unity; numbers that are not algebraic, like pi and e, are transcendental.
What it is for: Number theory, algebra and exact computation.
It can be find the minimal polynomial of a number; determine degree; distinguish algebraic and transcendental numbers; work with algebraic integers.
Distinguishing features
Countable set
Closed under arithmetic operations
Includes all rationals
Algebraic integers form a ring
What it looks like
Numbers expressed with radicals or as polynomial roots.
How it is recognised
Roots of integer polynomials
Degree equals minimal polynomial degree
Pi and e are transcendental
Related models
is a kind of - category
is contrasted with - complement
is studied in - field
forms - structure
In practice
Families and kinds
rational numbers
quadratic irrationals
algebraic integers
roots of unity
Perron and Pisot numbers
Standards and regulation
ISO 80000-2 notation
Failure modes and hazards
Assuming irrational means transcendental
Numerical approximation errors
Also called
Where this came from
wikidata · CC0 1.0
Also registered as vr.tr.algebraic-number
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Definition Is it algebraic.
Polynomials decide.
Test
Polynomial.
Test
Algebraicity.
- Is this number a root of a polynomial with integer coefficients? boundary
- Which polynomial? definition
Transcendental
Contrast.
Transcendental
Transcendence.
- Is the number known to be transcendental? provenance
- Or is its status open? boundary
Structure Degree.
Degree measures complexity.
Minimal polynomial
Degree.
Minimal polynomial
Minimal polynomial.
- What is its minimal polynomial and degree? measurement
- What are its conjugates? definition
Algebraic integers
Monic polynomials.
Algebraic integers
Algebraic integers.
- Is it an algebraic integer? boundary
- In which ring? definition
Computation Exact arithmetic.
Computers can handle them exactly.
Exact
Computer algebra.
Exact
Exact computation.
- How can it be represented exactly in software? action
- Which system supports it? provenance
Approximation
Numerics.
Approximation
Approximation.
- How accurate is the numerical approximation? measurement
- Could rounding mislead? boundary
Theory Number fields.
Algebraic numbers build fields.
Fields
Number fields.
Fields
Number fields.
- Which number field does it generate? definition
- What is its degree? measurement
History
Development.
History
History.
- How did the theory develop, including ideal numbers? provenance
- Which mathematicians contributed? provenance
What the second pass must settle
- Should algebraic integers be a separate entry?
- How should computer algebra be linked?
- How should number fields be linked?