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NumberTheory

  

NumberOfIrreduciblePolynomials

  

number of monic irreducible polynomials

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

NumberOfIrreduciblePolynomials(n, p)

NumberOfIrreduciblePolynomials(n, p, m)

Parameters

n

-

non-negative integer

p

-

power of prime number

m

-

(optional) positive integer; defaults to 1

Description

• 

The NumberOfIrreduciblePolynomials(n, p, m) command computes the number of monic irreducible univariate polynomials of degree n over a finite field of order pm.

• 

An explicit formula for this function is 1nd|nμdpmnd where the sum is over the divisors of n and μ is the Moebius function.

Examples

withNumberTheory:

NumberOfIrreduciblePolynomials3,5

40

(1)

NumberOfIrreduciblePolynomials1,24

16

(2)

The number of linear, quadratic, cubic, and quartics over GFp.

seqNumberOfIrreduciblePolynomialsn,p,n=1..4

p,12p212p,13p313p,14p414p2

(3)

The number of cubics over GFpm.

NumberOfIrreduciblePolynomials3,p,m

pm33pm3

(4)

Compatibility

• 

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

• 

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

See Also

NumberTheory