inert irreducibility function - Maple Help

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Irreduc - inert irreducibility function

Calling Sequence

Irreduc(a)

Irreduc(a, K)

Parameters

a

-

multivariate polynomial

K

-

RootOf

Description

• 

The Irreduc function is a placeholder for testing the irreducibility of the multivariate polynomial a. It is used in conjunction with mod and modp1.

• 

Formally, an element a of a commutative ring R is said to be "irreducible" if it is not zero, not a unit, and a=bc implies either b or c is a unit.

• 

In this context where R is the ring of polynomials over the integers mod p, which is a finite field, the units are the non-zero constant polynomials. Hence all constant polynomials are not irreducible by this definition.

• 

The call Irreduc(a) mod p returns true iff a is "irreducible" modulo p. The polynomial a must have rational coefficients or coefficients from a finite field specified by RootOf expressions.

• 

The call Irreduc(a, K) mod p returns true iff a is "irreducible" modulo p over the finite field defined by K, an algebraic extension of the integers mod p where K is a RootOf.

• 

The call modp1(Irreduc(a), p) returns true iff a is "irreducible" modulo p. The polynomial a must be in the modp1 representation.

Examples

Irreduc2mod7

false

(1)

Irreduc2x2+6x+6mod7

false

(2)

Irreducx4+x+1mod2

true

(3)

aliasα=RootOfx4+x+1:

Irreducx4+x+1,αmod2

false

(4)

Factorx4+x+1,αmod2

x+α+1α2+xx+αα2+x+1

(5)

See Also

AIrreduc, Factor, irreduc, isprime, mod, modp1, RootOf


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