Factorization of polynomials

Summary

Factorization of polynomials writes a polynomial as a product of irreducible polynomials. Over a field the result is unique up to order and constant factors, because $K[x]$ is a unique factorisation domain. The answer depends on the field: $x^2 + 1$ is irreducible over $\R$ but splits over $\C$. Computing factorizations is a central task of computer algebra, with different methods over finite fields, over $\Q$ and over number fields.