Valuation Theory
Valuation theory may be described as the study of divisibility (in commutative rings)
in its purest form, but that is only one aspect. The general formulation leads to the
introduction of topological concepts like completion, which provides a powerful
tool. It also emphasizes the parallel with the absolute value on the real and complex
numbers. After the initial definitions (in Section 9.1) we shall prove the essential
uniqueness of the absolute value on Rand C (in Section 9.2) and go on to describe
the p-adic numbers in Section 9.3 and integral elements in Section 9.4, before look-
ing at simple cases of the extension problem in Section 9.5.
9.1 Divisibility and Valuations
Let R be a (commutative) integral domain with field of fractions K; our object will be
to study the divisibility on R. We note that the divisibility can be defined on K as well
by writing for any a, b E KX, alb (a divides b) whenever b = ac for some c E R. This
relation is reflexive and transitive, thus it is a preordering of KX. Moreover, this
preordering is preserved by multiplication:
alb:::} adlbd for all a, b, d E KX ,
and so K may be regarded as a preordered group. It is not generally ordered, because
alb, bla need not imply a = b; but if we define
a '" b whenever alb and bla,
then we obtain an equivalence on KX. Its classes are just the classes of associated
elements of K X , and these classes form a partially ordered group r. If v(a) denotes
the class of a E K, then a 1-+ v(a) is a homomorphism from K to r. We shall write
the operation in r as addition and put v(a) .::: v(b) or v(b) ::: v(a) to indicate that
alb. Then the mapping v satisfies the conditions:
V.I v(a), v(b) > y:::} v(a + b) > y, for all y E r;
V.2 v(ab) = v(a) + v(b);
V.3 a E R {} v(a) ::: O.
307
,308 Valuation Theory
It is convenient to define v on the whole of K, including O. To achieve this we shall
introduce a symbol 00 satisfying 00 + y = 00, while y < 00 for all y E r. Ifwe now
put v(O) = 00, then V.I-V.3 hold for all a, bE K and y E r U {oo}. Of course this is
just a formal device, designed to ensure that our formulae hold without exception; in
defining v we can ignore v(O) since its value is prescribed.
Given a field K, it is clear that any function v from K to a partially ordered group
r, satisfying V.I, V.2, determines the subset R = {x E Klv(x ::: O}, which is easily
seen to be a subring, and V.3 holds for this R. So here we have another way of
describing R, which stresses the divisibility in R. In general this offers no advantage
over the usual description, but it suggests looking at rings for which r has a parti-
cularly simple form.
An important simplification is obtained by assuming r to be totally ordered. Then
V.I can be replaced by
V.I' v(a + b) ::: min {v(a), v(b)}.
In this case v is called a valuation on K and K is valuated. Thus a valuation on K is a
mapping v : K X ~ r to a totally ordered group r, together with v(O) = 00, satisfy-
ing V.I' and V.2. The ring R defined by V.3 is then called the valuation ring, or the
ring of valuation integers. It is easy to characterize valuation rings directly:
Proposition 9.1.1. Let K be a field. Then a subring V of K is a valuation ring in K if
and only if for any x E KX, either x E V or X-I E V.
Proof. Let v be a valuation on K and V = {x E Klv(x) ::: O} be the ring of integers.
Then for any x E K, v(x) + v(x- I ) = v(xx- I ) = v(l) = 0; hence either v(x) ::: 0 or
v(x- I ) ::: 0, so V satisfies the given conditions. Conversely, let V be a subring of K
satisfying this condition and denote by r the group of classes of associated elements,
defined as above. Then for any value Y E r, y = v(x) for some x E KX. Now either
x E V or X-I E V and accordingly y ::: 0 or -y ::: 0; this means that r is totally
ordered. •
Frequently the group r will be still further restricted. Thus if r = R, we speak of
a real-valued valuation. When r = Z, the simplest non-trivial case, we speak of a
principal valuation; this is sometimes called a discrete rank one valuation.
Every field has the trivial valuation, given by
v(x) = {0 if x :;to 0
00 if x = o.
Here are other examples, important in what follows:
Example 1. The p-adic valuation on Q. Every rational number can, up to sign, be
written uniquely as a product of powers of different primes. Fix a prime p; for
any a E QX let pV (where v E Z) be the exact power of p occurring in a and
define v(a) = v. This is easily seen to be a valuation, called the p-adic valuation.
For example, taking p = 3, we have v(lll) = I, v(lOj9) = -2. We remark that
,9.1 Divisibility and Valuations 309
this valuation is completely determined by its values on Z. This is a general property:
if K is the field of fractions of an integral domain A, then any valuation defined on A
has a unique extension to K (see Exercise 2).
Example 2. On k(x), the field of rational functions in x over a field k, a valuation may
be defined by writing any element cp of k(x) in reduced form: cp = f /g, where f, g are
polynomials in x over k which are coprime, and setting v( cp) = deg g - deg f. The
integers in this valuation are the rational functions remaining finite at 00. Other
valuations of k(x) are obtained by singling out an irreducible polynomial p over k
and defining the value of cp as the exponent of p in a complete factorization of cpo
In particular, for k = C (or any algebraically closed field) any irreducible polynomial
has the form x - ex, for some ex E C (up to a constant factor) and the integers for the
corresponding valuation are the rational functions that remain finite at x = ex. Thus
the valuations we have found correspond to the different places on the Riemann
sphere.
In what follows we shall mainly be concerned with principal valuations; although
some of the results will hold for the general case, we shall not always mention this
fact explicitly.
If v is a principal valuation on a field K, then the image v(KX) is a subgroup of Z.
It is either 0, when v is trivial, or it is of the form v(KX) = rZ for some r > 0, and
hence isomorphic to Z. In that case (l/r)v(x) is again a valuation, this time with
value group exactly Z. We call the valuation normalized if the value group is the
whole of Z and express the preceding statement by saying: every non-trivial (prin-
cipal) valuation can be normalized. Nevertheless, it is not always expedient to
normalize our valuations; e.g. a normalized valuation on K may be non-trivial on a
subfield F and yet not normalized on F.
We now turn to the consequences of the definition. As in every homomorphism of
groups, the neutral element is preserved, i.e. v( 1) = O. It follows that
v( - 1) = 0 and v( - x) = vex) for all x, (9.1.1)
for v( - 1) + v( - 1) = v(( - 1)2) = v(l) = 0, hence v( - 1) =0 and now
v( - x) = v( - 1) + vex) = vex).
The following relation will frequently be needed:
Lemma 9.1.2. For any x, y in a field with a valuation v,
vex - y) ::: min {vex), v(y)}, (9.1.2)
with equality unless vex) = v(y).
Proof. From V.I' we have vex - y) ::: min {vex), v( - y)} = min {vex), v(y)}, by
(9.1.1), and this proves (9.1.2). If vex) =f. v(y), say vex) > v(y), then
v(y) ::: min {vex), vex - y)} and vex - y) ::: v(y) by (9.1.2), hence vex - y) = v(y).
Similarly if v(y) > vex). •
, 310 Valuation Theory
By an easy induction we find that v( L~ ai) ::: min {v(al), ... , v(a n)}, and hence
we obtain, as in Lemma 9.1.2,
n
if L ai = 0, then v(ai) = v(aj) for some i :j:. j.
I
(9.1.3)
If we think of vex) as indicating the degree of divisibility of x by a certain prime,
property (9.1.2) is obvious. Later (in Section 9.2) we shall meet a less obvious
interpretation.
We have already seen that associated with every valuation there is a ring
v= {x E Klv(x) ::: O},
the ring of valuation integers. The set of all non-units in this ring is
I' = {x E Klv(x) > O};
clearly this is an ideal in V, by V.I', V.2, and it is maximal, since a proper ideal
cannot contain any units. Moreover, it is the unique maximal ideal of V. Hence
the quotient ring VII' is a field K, called the residue class field of the valuation v.
If v is the trivial valuation, then V = K, I' = 0 and the residue class field is just K
itself. Leaving this case aside, we may characterize the valuation rings of principal
valuations as follows:
Proposition 9.1.3. Let K be a field with a valuation v and valuation ring V. Then v is a
principal valuation if and only if V is a principal ideal domain, and in that case K
contains an element p such that
K X = (p) x u,
where (p) is the (multiplicative) cyclic group on p and U is the group of units of V
Thus every element a of K X has the form a = pnu, where n E Z and u E U, and if v
is normalized, then v(a) = n; in any case a E V if and only if n ::: o.
The element p is called a uniformizer or also a prime element of v; it is determined
up to a unit factor.
Proof. Let v be a principal valuation and take an element p of least positive value. By
taking v to be normalized we may assume that v(p) = 1; hence for any a E KX, if
v(a) = nEZ, then v(ap-n) = v(a) - nv(p) = 0, so ap-n = u E U, and we can
therefore write a = pnu, where n = v(a), u E U. It is clear that the representation
is unique once p has been chosen. Moreover, V is principal; all its ideals have the
form 0 or (pn), n ::: o.
Conversely, assume that V is principal; then its maximal ideal can be written in the
form (p). We claim that n(pn) = o. For if not, suppose that n(pn) = (q); then
q = anpn(a n E V) for all n::: 0, hence qp-I = anpn-I for all n::: 1, so qp-I E
n(pn) = (q), say qp-I = qb(b E V). Thus q(l - pb) = 0, but q:j:. 0, hence pb = 1, a
contradiction, and this shows that n(pn) = o. Now take a E KX; by what has been
proved, a ~ (pn + I) for some n. Choose the least such n; then a E (pn), so
Valuation theory may be described as the study of divisibility (in commutative rings)
in its purest form, but that is only one aspect. The general formulation leads to the
introduction of topological concepts like completion, which provides a powerful
tool. It also emphasizes the parallel with the absolute value on the real and complex
numbers. After the initial definitions (in Section 9.1) we shall prove the essential
uniqueness of the absolute value on Rand C (in Section 9.2) and go on to describe
the p-adic numbers in Section 9.3 and integral elements in Section 9.4, before look-
ing at simple cases of the extension problem in Section 9.5.
9.1 Divisibility and Valuations
Let R be a (commutative) integral domain with field of fractions K; our object will be
to study the divisibility on R. We note that the divisibility can be defined on K as well
by writing for any a, b E KX, alb (a divides b) whenever b = ac for some c E R. This
relation is reflexive and transitive, thus it is a preordering of KX. Moreover, this
preordering is preserved by multiplication:
alb:::} adlbd for all a, b, d E KX ,
and so K may be regarded as a preordered group. It is not generally ordered, because
alb, bla need not imply a = b; but if we define
a '" b whenever alb and bla,
then we obtain an equivalence on KX. Its classes are just the classes of associated
elements of K X , and these classes form a partially ordered group r. If v(a) denotes
the class of a E K, then a 1-+ v(a) is a homomorphism from K to r. We shall write
the operation in r as addition and put v(a) .::: v(b) or v(b) ::: v(a) to indicate that
alb. Then the mapping v satisfies the conditions:
V.I v(a), v(b) > y:::} v(a + b) > y, for all y E r;
V.2 v(ab) = v(a) + v(b);
V.3 a E R {} v(a) ::: O.
307
,308 Valuation Theory
It is convenient to define v on the whole of K, including O. To achieve this we shall
introduce a symbol 00 satisfying 00 + y = 00, while y < 00 for all y E r. Ifwe now
put v(O) = 00, then V.I-V.3 hold for all a, bE K and y E r U {oo}. Of course this is
just a formal device, designed to ensure that our formulae hold without exception; in
defining v we can ignore v(O) since its value is prescribed.
Given a field K, it is clear that any function v from K to a partially ordered group
r, satisfying V.I, V.2, determines the subset R = {x E Klv(x ::: O}, which is easily
seen to be a subring, and V.3 holds for this R. So here we have another way of
describing R, which stresses the divisibility in R. In general this offers no advantage
over the usual description, but it suggests looking at rings for which r has a parti-
cularly simple form.
An important simplification is obtained by assuming r to be totally ordered. Then
V.I can be replaced by
V.I' v(a + b) ::: min {v(a), v(b)}.
In this case v is called a valuation on K and K is valuated. Thus a valuation on K is a
mapping v : K X ~ r to a totally ordered group r, together with v(O) = 00, satisfy-
ing V.I' and V.2. The ring R defined by V.3 is then called the valuation ring, or the
ring of valuation integers. It is easy to characterize valuation rings directly:
Proposition 9.1.1. Let K be a field. Then a subring V of K is a valuation ring in K if
and only if for any x E KX, either x E V or X-I E V.
Proof. Let v be a valuation on K and V = {x E Klv(x) ::: O} be the ring of integers.
Then for any x E K, v(x) + v(x- I ) = v(xx- I ) = v(l) = 0; hence either v(x) ::: 0 or
v(x- I ) ::: 0, so V satisfies the given conditions. Conversely, let V be a subring of K
satisfying this condition and denote by r the group of classes of associated elements,
defined as above. Then for any value Y E r, y = v(x) for some x E KX. Now either
x E V or X-I E V and accordingly y ::: 0 or -y ::: 0; this means that r is totally
ordered. •
Frequently the group r will be still further restricted. Thus if r = R, we speak of
a real-valued valuation. When r = Z, the simplest non-trivial case, we speak of a
principal valuation; this is sometimes called a discrete rank one valuation.
Every field has the trivial valuation, given by
v(x) = {0 if x :;to 0
00 if x = o.
Here are other examples, important in what follows:
Example 1. The p-adic valuation on Q. Every rational number can, up to sign, be
written uniquely as a product of powers of different primes. Fix a prime p; for
any a E QX let pV (where v E Z) be the exact power of p occurring in a and
define v(a) = v. This is easily seen to be a valuation, called the p-adic valuation.
For example, taking p = 3, we have v(lll) = I, v(lOj9) = -2. We remark that
,9.1 Divisibility and Valuations 309
this valuation is completely determined by its values on Z. This is a general property:
if K is the field of fractions of an integral domain A, then any valuation defined on A
has a unique extension to K (see Exercise 2).
Example 2. On k(x), the field of rational functions in x over a field k, a valuation may
be defined by writing any element cp of k(x) in reduced form: cp = f /g, where f, g are
polynomials in x over k which are coprime, and setting v( cp) = deg g - deg f. The
integers in this valuation are the rational functions remaining finite at 00. Other
valuations of k(x) are obtained by singling out an irreducible polynomial p over k
and defining the value of cp as the exponent of p in a complete factorization of cpo
In particular, for k = C (or any algebraically closed field) any irreducible polynomial
has the form x - ex, for some ex E C (up to a constant factor) and the integers for the
corresponding valuation are the rational functions that remain finite at x = ex. Thus
the valuations we have found correspond to the different places on the Riemann
sphere.
In what follows we shall mainly be concerned with principal valuations; although
some of the results will hold for the general case, we shall not always mention this
fact explicitly.
If v is a principal valuation on a field K, then the image v(KX) is a subgroup of Z.
It is either 0, when v is trivial, or it is of the form v(KX) = rZ for some r > 0, and
hence isomorphic to Z. In that case (l/r)v(x) is again a valuation, this time with
value group exactly Z. We call the valuation normalized if the value group is the
whole of Z and express the preceding statement by saying: every non-trivial (prin-
cipal) valuation can be normalized. Nevertheless, it is not always expedient to
normalize our valuations; e.g. a normalized valuation on K may be non-trivial on a
subfield F and yet not normalized on F.
We now turn to the consequences of the definition. As in every homomorphism of
groups, the neutral element is preserved, i.e. v( 1) = O. It follows that
v( - 1) = 0 and v( - x) = vex) for all x, (9.1.1)
for v( - 1) + v( - 1) = v(( - 1)2) = v(l) = 0, hence v( - 1) =0 and now
v( - x) = v( - 1) + vex) = vex).
The following relation will frequently be needed:
Lemma 9.1.2. For any x, y in a field with a valuation v,
vex - y) ::: min {vex), v(y)}, (9.1.2)
with equality unless vex) = v(y).
Proof. From V.I' we have vex - y) ::: min {vex), v( - y)} = min {vex), v(y)}, by
(9.1.1), and this proves (9.1.2). If vex) =f. v(y), say vex) > v(y), then
v(y) ::: min {vex), vex - y)} and vex - y) ::: v(y) by (9.1.2), hence vex - y) = v(y).
Similarly if v(y) > vex). •
, 310 Valuation Theory
By an easy induction we find that v( L~ ai) ::: min {v(al), ... , v(a n)}, and hence
we obtain, as in Lemma 9.1.2,
n
if L ai = 0, then v(ai) = v(aj) for some i :j:. j.
I
(9.1.3)
If we think of vex) as indicating the degree of divisibility of x by a certain prime,
property (9.1.2) is obvious. Later (in Section 9.2) we shall meet a less obvious
interpretation.
We have already seen that associated with every valuation there is a ring
v= {x E Klv(x) ::: O},
the ring of valuation integers. The set of all non-units in this ring is
I' = {x E Klv(x) > O};
clearly this is an ideal in V, by V.I', V.2, and it is maximal, since a proper ideal
cannot contain any units. Moreover, it is the unique maximal ideal of V. Hence
the quotient ring VII' is a field K, called the residue class field of the valuation v.
If v is the trivial valuation, then V = K, I' = 0 and the residue class field is just K
itself. Leaving this case aside, we may characterize the valuation rings of principal
valuations as follows:
Proposition 9.1.3. Let K be a field with a valuation v and valuation ring V. Then v is a
principal valuation if and only if V is a principal ideal domain, and in that case K
contains an element p such that
K X = (p) x u,
where (p) is the (multiplicative) cyclic group on p and U is the group of units of V
Thus every element a of K X has the form a = pnu, where n E Z and u E U, and if v
is normalized, then v(a) = n; in any case a E V if and only if n ::: o.
The element p is called a uniformizer or also a prime element of v; it is determined
up to a unit factor.
Proof. Let v be a principal valuation and take an element p of least positive value. By
taking v to be normalized we may assume that v(p) = 1; hence for any a E KX, if
v(a) = nEZ, then v(ap-n) = v(a) - nv(p) = 0, so ap-n = u E U, and we can
therefore write a = pnu, where n = v(a), u E U. It is clear that the representation
is unique once p has been chosen. Moreover, V is principal; all its ideals have the
form 0 or (pn), n ::: o.
Conversely, assume that V is principal; then its maximal ideal can be written in the
form (p). We claim that n(pn) = o. For if not, suppose that n(pn) = (q); then
q = anpn(a n E V) for all n::: 0, hence qp-I = anpn-I for all n::: 1, so qp-I E
n(pn) = (q), say qp-I = qb(b E V). Thus q(l - pb) = 0, but q:j:. 0, hence pb = 1, a
contradiction, and this shows that n(pn) = o. Now take a E KX; by what has been
proved, a ~ (pn + I) for some n. Choose the least such n; then a E (pn), so
