8. Prove Theorem 2.46 as follows. (a) Show that the map : n k

8. Prove Theorem 2.46 as follows.

(a) Show that the map η: Ζ_n → Π^k_i=1 Ζ_ni, with

^def

η[a]_n = ([a]_n1, ..., [a]_nk)

is well-defined.

(2.34)

(b) For m_i ^def = n/n_i, show that there exists some m_i^-1 ∈ Z such that [m_i^-1]_ni is an inverse of [m_i]_ni in Z_ni^*.

(c) For c_i ^def = m_i ⋅ (m_i^-1 mod n_i) ∈ Z, consider the map ψ of type Π^k_i=1 Z_ni → Z_n

with

^def

ψ([a₁]_n1, ..., [a_k]_nk) ≡ [a₁ ⋅ c₁ + a₂ ⋅ c₂ + ... + a_k ⋅ c_k]_n.

Show that ψ is well-defined.

(d) Prove that η preserves the two-sided identity and group multiplication.

(e) Prove that ψ preserves the two-sided identity and group multiplication.

(f) Prove that η and ψ are mutually inverse functions.