Let (R, +R, *R) be a commutative ring with unity (with additive identity 0R ∈ R and multiplicative identity 1R ∈ R). Recall that R* denotes the subset of units, defined by R* = {u ∈ R: ∃v ∈ R satisfying u *R v = 1R}. Prove that R* satisfies the following properties: For each pair u, v ∈ R*, we have u *R v ∈ R*. For each u ∈ R*, any v ∈ R for which u *R v = 1R must satisfy v ∈ R*. 1R ∈ R*. These properties are summarized by saying that R* is an abelian group with respect to the binary operation *R; for this reason, we refer to R* as the group of units of R. Prove that Z[i√n] = {a + b√n: a, b ∈ Z} is a subring of R (Recall that this amounts to showing that it is closed under addition and multiplication, and that it contains 0 and 1). As discussed in class, this implies that Z[i√n] is a commutative ring with unity, with respect to the usual addition and multiplication of complex numbers.
(c) Deduce from (6), together with part (1) of (4), that the unit group Z[i√n]* is infinite (Hint: start by verifying that ∃V1 ∈ Z[i√n]).