16 1. Elementary Prime Number Theory, I

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Figure 1. The lattice Z + Zη sitting inside C. Here A = 2 so that D = −7.

and that (iii) implies (i). To continue we need some preliminary results

on the arithmetic of the rings Z[(−1 +

√

D)/2]. These will be familiar to

students of algebraic number theory, but we include full proofs for the sake

of completeness.

Let A ≥ 2 be an integer, and fix a complex root η of x√+

2

x + A, so

that (for an appropriate choice of the square root) η = (−1 + D)/2. Since

η2

= −η − A, it follows that

Z[η] = Z + Zη = {x + yη : x, y ∈ Z}.

For α ∈ Z[η], we denote its complex conjugate by α. Observe that η =

−1 − η; consequently, Z[η] is closed under complex-conjugation. We define

the norm of the element α = x + yη ∈ Z[η] by

N (α) : =

|α|2

= αα =

x2

− xy +

Ay2.

Notice that the norm of α ∈ Z[η] is always an integer and is positive when-

ever α = 0. Moreover, since the complex absolute value is multiplicative, it

is immediate that

N (αβ) = N (α) · N (β) for all α, β ∈ Z[η].

We now recall the requisite definitions from ring theory: If α, β ∈ Z[η],

we say that α divides β if β = αγ for some γ ∈ Z[η]. A nonzero element

α ∈ Z[η] is called a unit if α divides 1. A nonunit element α ∈ Z[η] is

irreducible if whenever α = βγ with β, γ ∈ Z[η], then either β is a unit or

γ is a unit. Finally, π ∈ Z[η] is called prime if whenever π divides βγ for

β, γ ∈ Z[η], then either π divides β or π divides γ.