Exercise 29, § 1 Commutative Algebra, IX Let AAA be a ring. Let n∈Jn \in Jn∈J, ppp a prime number, and a∈AJa \in A^Ja∈AJ. Establish the relation Φpn(a)=Φn(ap)+pwΦm(fpw(a)), where w=wp(pn) and pn=pwm. We therefore have \Phi_{pn}(a) = \Phi_n(a^p) + p^w \Phi_m(f_{p^w}(a)), \text{ where } w = w_p(pn) \text{ and } pn = p^w m. \text{ We therefore have} Φpn(a)=Φn(ap)+pwΦm(fpw(a)), where w=wp(pn) and pn=pwm. We therefore have Φpn(a)≡Φn(ap)mod .pwp(pn)A,and in particularΦpn(a)≡Φn(a)pmod .pA. \Phi_{pn}(a) \equiv \Phi_n(a^p) \mod. p^{w_p(pn)}A, \quad \text{and in particular} \quad \Phi_{pn}(a) \equiv \Phi_n(a)^p \mod. pA . Φpn(a)≡Φn(ap)mod.pwp(pn)A,and in particularΦpn(a)≡Φn(a)pmod.pA. Solution No solution has been written yet.