Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors. For example, the polynomial ${\displaystyle x^{2}-2}$ does not factor over the rationals; its splitting field is the smallest extension of the rationals containing ${\displaystyle {\sqrt {2}}}$ and ${\displaystyle -{\sqrt {2}}}$.