Question

What is an irreducible polynomial?

Answer 1

An irreducible polynomial is one that cannot be factored into simpler (lower degree) polynomials using the kind of coefficients you are allowed to use, or is not factorisable at all.

Explanation:

Polynomials in a single variable

`x^2-2` is irreducible over `QQ`. It has no simpler factors with rational coefficients.

`x^2+1` is irreducible over `RR`. It has no simpler factors with Real coefficients.

The only polynomials in a single variable that are irreducible over `CC` are linear ones.

Polynomials in more than one variable

If you are given a polynomial in two variables with all terms of the same degree, e.g. `ax^2+bxy+cy^2`, then you can factor it with the same coefficients you would use for `ax^2+bx+c`.

If it is not homogeneous then it may not be possible to factor it. For example, `x^2+xy+y+1` is irreducible.