Built-in Mathematica Symbol

Decompose

Decompose[poly, x]
decomposes a polynomial, if possible, into a composition of simpler polynomials.

Decompose gives a list of the polynomials P_i which can be composed as to give the original polynomial.

Represent a polynomial as a composition of polynomials:

A composition of more than two polynomials:

No decomposition:

A polynomial with symbolic coefficients:

Decompose a polynomial over integers modulo 2:

Solve some polynomial equations of degrees higher than 4 in terms of radicals:

Solve

by solving

and then

etc:

Check that these indeed are the roots of f:

Mathematica solvers use Decompose automatically:

Composition of polynomials given by Decompose gives the original polynomial:

Use Fold to compose the polynomials:

Use Expand to show that the result is equal to f:

Use Factor to represent a polynomial as a product of irreducible factors:

f can be factored but not decomposed; g can be decomposed but not factored:

Decompose ignores possible decompositions with inner polynomials that are linear: