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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

 > $\mathrm{Irreduc}\left(2\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}7$
 ${\mathrm{false}}$ (1)
 > $\mathrm{Irreduc}\left(2{x}^{2}+6x+6\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}7$
 ${\mathrm{false}}$ (2)
 > $\mathrm{Irreduc}\left({x}^{4}+x+1\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 ${\mathrm{true}}$ (3)
 > $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({x}^{4}+x+1\right)\right):$
 > $\mathrm{Irreduc}\left({x}^{4}+x+1,\mathrm{\alpha }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 ${\mathrm{false}}$ (4)
 > $\mathrm{Factor}\left({x}^{4}+x+1,\mathrm{\alpha }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}2$
 $\left({{\mathrm{\alpha }}}^{{2}}{+}{x}{+}{1}\right){}\left({x}{+}{\mathrm{\alpha }}{+}{1}\right){}\left({{\mathrm{\alpha }}}^{{2}}{+}{x}\right){}\left({x}{+}{\mathrm{\alpha }}\right)$ (5)