16 1. Elementary Prime Number Theory, I

Figure 1. The lattice Z + 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√+
x + A, so
that (for an appropriate choice of the square root) η = (−1 + D)/2. Since
= −η A, it follows that
Z[η] = Z + = {x + : 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 + Z[η] by
N (α) : =
= αα =
xy +
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.
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