AlgebraicUnitQ

AlgebraicUnitQ[a]

yields True if a is an algebraic unit, and yields False otherwise.

Details

  • AlgebraicUnitQ is typically used to test whether a number is an algebraic unity or not.
  • An algebraic unit a is a number for which both a and 1/a are algebraic integers.
  • AlgebraicUnitQ[a] returns False unless a is manifestly an algebraic unit.

Examples

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Basic Examples  (2)

Test whether a number is an algebraic unit:

The number is not an algebraic unit:

Scope  (4)

AlgebraicUnitQ works over integers:

Real numbers:

Complex numbers:

Transcendental numbers:

Radical expressions:

Root objects:

AlgebraicNumber objects:

AlgebraicUnitQ threads over lists:

Applications  (6)

Basic Applications  (1)

Generate random algebraic units:

Plot algebraic units:

Number Theory  (5)

Find integers that are algebraic units:

All roots of unity are algebraic units:

An algebraic unit has norm or :

Representatives of norm in :

It can be represented in terms of the representative a by multiplying by a unit:

Use the roots of unity to find Cyclotomic polynomials:

Properties & Relations  (7)

An algebraic unit and its reciprocal are algebraic integers:

The reciprocal is an algebraic unit:

The only integers that are both algebraic integers and algebraic units are and :

An algebraic unit raised to a power is an algebraic unit:

Algebraics represents the domain of all algebraic numbers, including algebraic units:

An algebraic unit has norm or :

Use MinimalPolynomial to find the minimal polynomial of an algebraic unit:

Now find the minimal polynomial of its reciprocal:

Fundamental units of a number field are algebraic units:

Possible Issues  (1)

In some cases it is not known whether the number is an algebraic unit:

Introduced in 2007
 (6.0)