 It is common to construct an extension field of a given field "K" as a quotient ring of the polynomial ring "K"["X"] in order to "create" a root for a given polynomial "f"("X").Suppose for instance that "K" does not contain any element "x" with "x"2 = −1.Then the polynomial formula_25 is irreducible in "K"["X"], consequently the ideal generated by this polynomial is maximal, and formula_26 is an extension field of "K" which "does" contain an element whose square is −1 (namely the residue class of "X").