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NumberTheory

  

CyclotomicPolynomial

  

minimal polynomials of primitive roots of unity with rational coefficients

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

CyclotomicPolynomial(n, x)

 

Phi(n, x)

Φn

Parameters

n

-

positive integer

x

-

name

Description

• 

The CyclotomicPolynomial('n', 'x') command computes the nth cyclotomic polynomial in x.

• 

The roots of the nth cyclotomic polynomial are exactly the nth primitive roots of unity.

• 

The degree of the nth cyclotomic polynomial is given by Euler's totient function, NumberTheory[Totient].

• 

Phi is an alias for CyclotomicPolynomial.

• 

You can enter the command Phi using either the 1-D or 2-D calling sequence. For example, Phi(8, x) is equivalent to Φ8,x.

Examples

withNumberTheory:

CyclotomicPolynomial1,x

x1

(1)

Φ2,x

x+1

(2)

The one hundred and fifth cyclotomic polynomial is the first with a coefficient greater than 1.

CyclotomicPolynomial105,x

x48+x47+x46x43x422x41x40x39+x36+x35+x34+x33+x32+x31x28x26x24x22x20+x17+x16+x15+x14+x13+x12x9x82x7x6x5+x2+x+1

(3)

Totient105

48

(4)

pCyclotomicPolynomial7,x

px6+x5+x4+x3+x2+x+1

(5)

rsolvep=0,x

rcos2π7+Isin2π7,cos3π7+Isin3π7,cosπ7+Isinπ7,cosπ7Isinπ7,cos3π7Isin3π7,cos2π7Isin2π7

(6)

plots:-complexplotr,style=point

Compatibility

• 

The NumberTheory[CyclotomicPolynomial] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

NumberTheory

NumberTheory[IsCyclotomicPolynomial]

NumberTheory[Totient]