Hensel's lemma the value of "c" mod 27: if "c" ≡ 1 mod 27 then use "a" = 1, if "c" ≡ 10 mod 27 then use "a" = 4 (since 4 is a root of "f"("x") mod 27), and if "c" ≡ 19 mod 27 then use "a" = 7. (It is not true that every "c" ≡ 1 mod 3 is a 3-adic cube, e.g., 4 is not a 3-adic cube since it is not a cube mod 9.) In a similar way, after some preliminary work Hensel's lemma can be used to show that for any "odd" prime number
Principal root of unity Principal root of unity In mathematics, a principal "n"-th root of unity (where "n" is a positive integer) of a ring is an element formula_1 satisfying the equations In an integral domain, every primitive "n"-th root of unity is also a principal formula_3-th root of unity. In any ring, if formula_3 is a power of formula_5, then any formula_6-th root of formula_7 is a principal formula_3-th root of unity. A non-example is formula_9 in the ring of integers modulo formula_10; while formula_11 and thus formula_9 is a cube root of unity, formula_13 meaning that it is not a "principal" cube