Hensel’s Lemma And Hensel Lift Over Z₃â†'Z₉

Since Wan Zhexian built the Hensel’s lemma and Hensel lift of Z2[x]over Z4[x] and gave the structure of Galois ringGR(4m). The research of Hensel’s lemma and Hensel lift become an important topic in algebraic number theory and computer field, and also become an important tool to research and construction linear codes in a ring.A local ring is a ring that has a unique maximal ideal. If R is a local ring, the polynomial ring R[x]may have the unique factorization property.Z9is a local ring and the polynomial xn-1is the product of basic irreducible polynomials. This paper fully draws on the Hensel lift of xn-1over Z2â†'Z4, and build the Hensel lift of xn-1over Z3â†'Z9has been built. By proving of the Hensel lemma an algorithm is presented to compute the Hensel lift. And the ideals of Z9/(xn-1)is given at the same time, where(n,3)=1.