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58 Appendix to Chapter 2

(b) For an integral domain R such that every nonzero element is a product of irre-
ducible elements the following are equivalent:
(1) R is factorial.
(2) If p is an irreducible dividing ab, then p divides either a or b.
(3) If p is an irreducible, then Rp is a prime ideal.

(1.5) Example. Every principal ring is a factorial ring for example the integers Z,
the Gaussian integers Z[i], the Jacobi integers Z[ρ], where ρ = exp(2πi/3),and
k[X], where k is a field.

For R a factorial ring with field of fractions F and an irreducible p in R, each
r
nonzero x in F can be represented as x = p (a/b), where a, b are in R and not
divisible by p and r is an integer. Moreover, r and a/b are unique. We define the
∗
order function ord p : F → Z at the irreducible p by the relation ord p (x) = r.
Observe that it has the following properties:
ord p (xy) = ord p (x) + ord p (y),
ord p (x + y) ≥ min{ord p (x), ord p (y)}.

Moreover, any nonzero a in F is in R if and only if ord p (a) ≥ 0 for all irreducibles
p in R. This leads to the following definition.

(1.6) Definition. A (discrete) valuation on a field F is a function v : F → Z such
that
v(xy) = v(x) + v(y) and v(x + y) ≥ min{v(x), v(y)},

where x, y are in F. By convention we set v(0) =+∞ so that the case x + y = 0is
covered in the second relation. The value group of v is defined to be the image of v
in Z.

As for some elementary properties of valuations, observe that

x n
v = v(x) − v(y), v(1) = v(−1) = 0, v(x ) = nv(x),
y
and if v(x)<v(y), then it follows that v(x) = v(x + y). To see this, we have
v(x) ≤ v(x + y) from the definition, and from x = (x + y) + (−y), it follows that
v(x + y) ≤ v(x) too since v(y) = v(−y).
For a prime number p in Z the associated valuation ord p on Q is called the p-
adic valuation, and up to an integral multiple every valuation on the field of rational
numbers is a p-adic valuation for some p.
To each valuation we can associate a principal, hence factorial, ring with exactly
one irreducible up to multiplication by units.

(1.7) Definition. Let v be a valuation on a field F. The valuation ring R v is the set
of all x in F with v(x) ≥ 0.