4. Prove Theorem 2.35 as follows. (a) Show that and are well-defined. (b) Show that...

4. Prove Theorem 2.35 as follows.

(a) Show that η and ψ are well-defined.

(b) Show that η preserves two-sided identities, that is, η[1] = [0]¢(n).

(c) Show that ψ preserves two-sided identities, that is, ψ[0]_φ(n) = [1]_η.

(d) Show that η preserves group multiplication, that is,

η([x]_η *_η [y]_η) = η[x]_η +_φ(η) η[y]_η.

What equation must you show for the logarithm?

(e) Show that ψ preserves group multiplication, that is,

ψ([11]_φ(n) +_φ(n) [12]_φ(n)) = ψ[11]_φ(n) *_η ψ[12]_φ(n).

(f) Show that η(ψ[t]_φ(n)) = [t]_φ(n) for all [t]_φ(n)∈ Ζφ(π).

(g) Show that ψ(η[x]_η) = [x]_η for all [x]_η∈ Ζ_η.