Valuations, Metrics and Completions

Number Fields and Quadratic Fields

A.3 Valuations, Metrics and Completions

We now consider discrete valuations and metrics on an arbitrary field :F. A map v : :F --+ Z ^{U CX>} is called a discrete valuation of :F if:

1. v maps yx onto Z.

2. v(O) = CX>.

3. v(x

+

y) ~ min(v(x), v(y)) with strict inequality if v(x)

=/-

v(y).

A map

I 1 :

:F--+ R is called a metric of :F if for all x, y E :F:

1.

I

x I~ 0 and

I

x

I=

0 ~ x

=

0.

2.

I

xy

1=1

x

II

Y

I·

3. There is a constant C E R such that

11 +

^x

Is

^C if

I

^x

Is

1.

We shall say that two metrics

I

¹11

I b

on :F are equivalent if they define the same topology on :F. This is so if and only if there is a c

>

0 such that for all x E :F,

I

x l1

=I

x

12·

Every valuation is equivalent to one where we can take C

=

2 in Axiom 3. This gives us the usual triangle inequality:

I

x

+

y

I

S

I

x

I + I

Y

I·

A metric is discrete if there is a 8

>

0 such that 1 - 8

< I

x

I <

1

+

8 implies that

I

^x

I =

1. A metric is non-archimedean if one can take C = 1 in Axiom 3; this is so if

I

x

+

y

I S

max(l x

I, I

^Y

1).

APPENDIX A. NUMBER FIELDS AND QUADRATIC FIELDS 97 Now we focus on a number field :F. For a prime ideal p of S, one can consider the p-adic valuation vP on :F. Assume that N(p) = q = pf. If a E :Fx, we let

vP(a) = exact power of pin the factorization of (a).

Define vp(O) = oo. This makes v"' a discrete valuation. We can also define a p-adic metric on :F:

This defines a discrete non-archimedean metric.

It is known that every field on which a metric is defined can be embedded in its completion, which is a (unique) minimal field that is complete with respect to this metric. Denote by :Fp the g:J-adic completion of :F with respect to the given p-adic metric. The ring of integers of :F"', denoted by Sp, is defined by

a set which is in fact a discrete valuation ring, with fraction field :Fp· We shall denote also by p the maximal ideal of Sp, which consists of those ^a E Sp with absolute value strictly less than 1.

:Fp is a finite extension of Qp, the p-adic completion of Q under the usual p-adic metric. We shall only be interested in local fields which are completions of a number field, so we shall henceforth symbolize a (non-archimidean) local field by :Fp.

We shall denote by 1r"' a uniformizer for S"', that is, g:J = ( 1r "') . The units of S"' are those elements with absolute value exactly 1. The residue field kp = Spj(1rp) is an extension of Zjp of degree

f.

The canonical map Sp - t kp will be denoted by

-

_{, I.e.,}

^.

_r - t r

-

.

We can also consider field embeddings oo;

Each oo; defines an archimedean metric:

I

^a

loci

⁼

I

oo;(a)

I

:F-tC, i = l, ... ,n = [:F : Q].

APPENDIX A. NUMBER FIELDS AND QUADRATIC FIELDS 98

where the

I I

on the right side above is defined by:

for x,y E R,

I xI=

max (x,-x),

I

x+iy

I=

)x²

+y

²

The completion of :F that such a metric defines is isomorphic to either R or C.

We say that oo; is a real embedding if the completion it defines is R, and non-real otherwise. Non-real embeddings come in complex conjugate pairs oo; and oo; = cooo;, where c is complex conjugation. Thus if r is the number of real embeddings of :F and s is half the number of non-real embeddings, then n = r

+

2s. A number field :F is said to be totally real if every embedding oo; of :F yields R as its completion.

We can also speak of totally positive elements of :For S, which are those a such that oo;(a)

>

0 for every embedding oo; : :F--+ R. We denote these sets by :F>>O and S>>O· Similarly we can speak of totally negative elements of :For S.

We denote by M:r the set of inequivalent metrics on :F. The non-archimedean me tries will be denoted by

M]:.

and the archimedean ones by

M?.

It is known that these two classes are in 1-1 correspondence with the distinct primes in S and the distinct embeddings of :F in C, respectively, the correspondence constructed as above. Hence we shall also refer to them as the finite primes, symbolically g:J

<

oo, and infinite primes, respectively. As usual, the phrase "for almost all g:J" means "for all but finitely many g:J."

We denote by A:r the ring of adeles of :F. This consists of all vectors (aP)PEM.r where aP E Fp and aP E SP for almost all p. Addition and multiplication is defined componentwise. We have A:r = Ac; x A~, where Ac; are those adeles with 1 at the finite primes, and A~ those with 1 at the infinite primes. The units of A:r, denoted by I:r, is called the idele group of :F. :F and :Fx are embedded diagonally in A:r and I.r, respectively. These are called the principal adeles and ideles, respectively.

Dirichlet's Unit Theorem states that for any number field :F, the units, U, of the ring of integers S is a finitely generated abelian group. The torsion part of U is

APPENDIX A. NUMBER FIELDS AND QUADRATIC FIELDS 99 the cyclic group of roots of unity in :F. The torsion-free part of U is generated by r

+

^{s -} 1 elements of U.

The focus of this thesis is totally real quadratic fields. Henceforth, unless oth- erwise specified, m will always be a positive squarefree rational integer, F = Q(

Vffi)

and R the ring of integers ofF, U its units. From the above, the torsion free part of U is generated by a single element u

>

1, which will be called the fundamental unit of R. In other words, every unit is of the form ±uk for some k E Z. We shall see in Appendix E that u can be effectively computed, and analytic formulas are available to effectively compute h(F), based only on m and u.