Download Absolute Values for Rational Numbers and More Definition: A

Absolute Values for Rational Numbers and More
Definition: A function | · | from Q to itself is called an absolute value if it satisfies the
following:
i) |a| ≥ 0, with equality if and only if a = 0,
ii) |ab| = |a||b| for all a, b ∈ Q,
iii) |a + b| ≤ |a| + |b|.
The last condition here is called the triangle inequality. If the stronger condition
iv) |a + b| ≤ max{|a|, |b|}
holds, we say the absolute value is non-archimedean. If not, the absolute value is called archimedean.
Typically one allows the absolute value to take non-negative real values. There really isn’t
any loss of generality here, though (see the comment before Lemma 8). Our definition also has the
advantage of not assuming the existence of R, an advantage which will be apparent very soon.
Examples: The usual absolute value, which we’ll denote | · |∞ , is an archimedean absolute
value on Q. For any prime number p, the p-adic absolute value defined by |a|p = p−ordp (a) is
a non-archimedean absolute value. Here ordp (a) is the exponent of p that occurs in the unique
factorazation of a into a product of primes (take it to be ∞ if a = 0). For example, ord2 (18) = 1
and ord3 (18) = 2 since 18 = 21 32 . Futher, ordp (18) = 0 for all other primes p 6= 2, 3.
It turns out (Ostrowski’s Theorem) that the above examples are essentially all the absolute
values on Q. (Here “essentially” refers to the topologies produced.) For this reason, one often refers
to the “prime at infinity” as giving the usual archimedean absolute value. We’ll use v to denote
either a prime number or the prime at infinity; a v-adic absolute value is either | · |∞ or one of the
p-adic absolute values above.
Definition: Let | · | be an absolute value on Q. A Cauchy sequence is a sequence {an } of
rational numbers such that, for all ε > 0, there is an N ∈ N where |an − am | < ε for all n, m ≥ N .
Examples:
1) For a fixed c ∈ Q, the constant sequence an = c for all n is a Cauchy sequence for all | · |v . The
special case where c = 0 is called the null sequence.
2) The sequence given by an =
10n −1
10n
is a Cauchy sequence for the usual absolute value, but not
1
for any of the p-adic absolute values.
3) The sequence given by an = 2n is a Cauchy sequence for the 2-adic absolute value, but not for
any other v-adic absolute value.
Lemma 1: Suppose {an } and {bn } are Cauchy sequences. Then so are the sequences given
by cn = an + bn and dn = an bn .
Proof: Let ε > 0 and let M, N ∈ N with |am − an | < ε/2 for all m, n ≥ M and |bm − bn | < ε/2
for all m, n ≥ N . Then for all m, n ≥ max{M, N } we have
|cm − cn | = |am − an + bm − bn | ≤ |am − an | + |bm − bn | < ε.
Let M0 , N0 ∈ N with |am − an | < 1 for all m, n ≥ M0 and |bm − bn | < 1 for all m, n ≥ N0 .
Then by the triangle inequality |am | < |aM0 |+1 for all m ≥ M0 and |bm | ≤ |bN0 |+1 for all m ≥ N0 .
Let M, N ∈ N with |am − an | < ε/2(|bN0 | + 1) for all m, n ≥ M and |bm − bn | < ε/2(|aM0 | + 1) for
all m, n ≥ N . Then for all m, n ≥ max{M0 , N0 , M, N } we have
|dm − dn | = |(am − an )bm + (bm − bn )an | ≤ |am − an ||bm | + |bm − bn ||an | < ε.
Definition: Let | · | be an absolute value on Q and let c ∈ Q. A sequence {an } is said to
converge to c in the absolute value | · |, and we write an → c, if for every ε > 0 there is an N ∈ N
such that |an − c| < ε for all n ≥ N .
Lemma 2: If a sequence {an } converges in an absolute value, then it is a Cauchy sequence
for that absolute value.
Proof: Suppose an → c and let ε > 0. Get an N ∈ N such that |an − c| < ε/2 for all n ≥ N .
Then by the triangle inequality
|an − am | = |(an − c) − (am − c)| ≤ |an − c| + |am − c| < ε
for all n, m ≥ N .
In the examples of Cauchy sequences above,
10n −1
10n
→ 1 in the usual absolute value and 2n → 0
in the 2-adic absolute value. Such Cauchy sequences are actually relatively uncommon, it turns
out. Specifically, “most” Cauchy sequences don’t converge.
Definition: Two Cauchy sequences {an } and {bn } of rational numbers are called equivalent,
and we write {an } ∼ {bn }, if an − bn → 0.
2
Lemma 3: This is an equivalence relation.
Proof: For any sequence {an } of rational numbers, an − an → 0, so that {an } ∼ {an } for any
Cauchy sequence {an }.
Suppose {an } ∼ {bn } and let > 0. Then for some N ∈ N, |an − bn | < for all n ≥ N . Thus
|bn − an | < for n ≥ N and bn − an → 0. (Note that | − 1| = 1 from part ii of the definition of
absolute value.) In other words, {bn } ∼ {an }.
Suppose {an } ∼ {bn } and {bn } ∼ {cn } and let > 0. Then for some N ∈ N, |an − bn | < /2
for all n ≥ N and for some M ∈ N, |bn − cn | < /2 for all n ≥ M . This implies that
|an − cn | = |an − bn + bn − cn | ≤ |an − bn | + |bn − cn | <
+ =
2 2
for all n ≥ max{N, M }. Thus {an − cn } → 0 and {an } ∼ {cn }.
Definition/Notation: The real numbers R is the set of equivalence classes of Cauchy sequences of rational numbers for the archimedean absolute value; we also write Q∞ for the real
numbers. The p-adic numbers Qp is the set of equivalence classes of Cauchy sequences for the
p-adic absolute value. For a Cauchy sequence {an } we’ll write [{an }] to denote the equivalence
class of Cauchy sequences {bn } with {an } ∼ {bn }. We view Q as a subset of Qv by identifying the
rational number c with the sequence given by an = c for all n.
Definition: Define addition and multiplication in Qv by
[{an }] + [{bn }] = [{an + bn }]
and
[{an }] · [{bn }] = [{an · bn }].
Lemma 4: These operations are well-defined, i.e., depend only on the equivalence classes and
not the particular elements of the equivalence classes used.
Proof: Suppose {an } ∼ {a0n } and {bn } ∼ {b0n } are all Cauchy sequences. By Lemma 2, both
{an + bn } and {an bn } are Cauchy sequences.
Let > 0. Then for some N1 , M1 ∈ N, |an − a0n | < /2 for all n ≥ N1 and |bn − b0n | < /2 for
all n ≥ M1 . This implies that
|(an + bn ) − (a0n + b0n )| = |(an − a0n ) + (bn − b0n )| ≤ |an − a0n | + |bn − b0n | <
3
+ =
2 2
for all n ≥ max{N1 , M1 }. Thus, {an + bn } ∼ {a0n + b0n } and addition is well-defined.
By the proof of Lemma 2, all four sequences {an }, {a0n }, {bn } and {b0n } are bounded. Thus,
there is a B ≥ 1 such that |an |, |a0n |, |bn |, |b0n | ≤ B for all n ∈ N. For some N2 , M2 ∈ N we have
|an − a0n | <
2B
for all n ≥ N2 and |bn − b0n | <
2B
for all n ≥ M2 . This implies that
|an bn − a0n b0n | = |(an − a0n )bn + a0n (bn − b0n )|
≤ |an − a0n ||bn | + |a0n ||bn − b0n |
<
·B+
·B
2B
2B
=
for all n ≥ max{N2 , M2 }. Thus, |an bn − a0n b0n | < for all n ≥ max{N2 , M2 } and {an bn } ∼ {a0n b0n }.
This shows that multiplication is well-defined.
Theorem 1: For any v, Qv is a field containing (an isomorphic copy of) Q.
Proof: The only thing not immediate here is the existence of multiplicative inverses. Suppose
that [{an }] is not equivalent to the null sequence. Then for some > 0 there are infinitely many
n ∈ N such that |an | ≥ . Since {an } is a Cauchy sequence, there is an N ∈ N such that
|an − am | < /2 for all n, m ≥ N . Since there must be an n0 ≥ N such that |an0 | ≥ , we have
|am | ≥ |an0 | − |an0 − am | > −
=
2
2
for all m ≥ N . In particular, am 6= 0 for m ≥ N . Further, it is not difficult to see that the sequence
{bn } defined by
bn =
aN
if n ≤ N ,
an
if n ≥ N
is a Cauchy sequence equivalent to {an }. Thus, we may assume without loss of generality that
an 6= 0 for all n ∈ N. Then easily [{a−1
n }] is a multiplicative inverse for [{an }].
Next, we extend our v-adic absolute values to Qv .
Definition: For [{an }] ∈ Qv , set [{an }]v = [{|an |v }] ∈ R
Lemma 5: This is well-defined, i.e., {|an |v } is a Cauchy sequence for the archimedean absolute
value and {|an |v } ∼ {|a0n |v } whenever {an } and {a0n } are equivalent Cauchy sequences for the v-adic
absolute value.
Proof: Let > 0. Then for some N ∈ N we have |an − am |v < for all n, m ≥ N . By the
triangle inequality, |an |v − |am |v ≤ |an − am |v and |am |v − |an |v ≤ |am − an |v . As remarked above,
4
|an −am |v = |am −an |v , so that |an |v −|am |v ∞ ≤ |an −am |v . This implies that |an |v −|am |v ∞ < for all n, m ≥ N and {|an |v } is a Cauchy sequence for the archimedean absolute value.
Suppose {a0n } ∼ {an } and let > 0. Then for some N ∈ N, |a0n − an |v < for all n ≥ N .
As above, this implies that |a0n |v − |an |v ∞ < for all n ≥ N , so that {|a0n |v } and {|an |v } are
equivalent Cauchy sequences for the archimedean absolute value.
In order to show these extensions are “absolute values,” we first need to discuss the archimedean
case.
Definition: We say a real number a is greater than zero (or positive) and write a > 0 if there
is an N ∈ N and an > 0 such that an ≥ for all n ≥ N , where {an } is a Cauchy sequence in the
equivalence class a.
Lemma 6: This is well-defined.
Proof: Suppose {a0n } and {an } are equivalent Cauchy sequences for the archimedean absolute
value. Suppose further that there is some N ∈ N and an > 0 such that an ≥ for all n ≥ N . There
is an M ∈ N such that |a0n − an | < /2 for all n ≥ M . This implies that |a0n | ≥ |an | − |a0n − an | >
−
2
=
2
for all n ≥ max{N, M }.
Definition: We say a real number a is greater than a real number b, and write a > b, if
a − b > 0.
Suppose a = [{an }] and b = [{bn }], where {an } and {bn } are Cauchy sequences for the
archimedean absolute value. Then a > b means there is an ε > 0 and N ∈ N with an ≥ bn + ε for
all n ≥ N . This implies that an > bn for all n ≥ N . Note, however, that an > bn for all n greater
than or equal to some N ∈ N does not imply that a > b (though it does imply that a ≥ b).
Lemma 7: The real numbers are totally ordered by | · |. In other words, for a, b ∈ R with
a 6= b, either a > b or b > a.
Proof: Write a = [{an }] and b = [{bn }] as above and suppose that a 6> b and b 6> a.
Let ε > 0. By Lemma 2, {an − bn } is a Cauchy sequence, so there is an N ∈ N such that
|(am − bm ) − (an − bn )| < ε for all m, n ≥ N . Suppose m ≥ N . If am − bm ≥ 0, then since a 6> b
there are infinitely many n ≥ N with an − bn ≤ 0. For such an n, we see by the above inequality
that am − bm < ε. Similarly, if am − bm ≤ 0, using b 6> a we get am − bm > −ε. Thus, |am − bm | < ε
for all m ≥ N and a − b = 0.
5
Theorem 2: As defined above, for any v the function | · |v on Qv is an absolute value. In
other words, the following three properties hold:
i) |a|v ≥ 0, with equality if and only if a = 0 for all a ∈ Qv ,
ii) |ab|v = |a|v |b|v for all a, b ∈ Qv ,
iii) |a + b|v ≤ |a|v + |b|v for all a, b ∈ Qv .
Moreover, we have
iv) |a + b|v ≤ max{|a|v , |b|v } for all a, b ∈ Qv
if v 6= ∞. That is, | · |v is non-archimedean for v 6= ∞.
Proof: Clearly |a|v ≥ 0 and |0|v = 0 (though one should be careful here as the first “0” is an
element of Qv while the second is an element of R = Q∞ ). Suppose that a 6= 0. Writing a = [{an }]
where {an } is a Cauchy sequence for the v-adic absolute value, there must be an > 0 such that
|an |v ≥ for infinitely many n ∈ N. Let N ∈ N be such that |an − am | < /2 for all n, m ≥ N .
Since there is an n0 ≥ N with |an0 |v ≥ , we have |am |v ≥ |an0 |v − |an0 − am |v > − /2 = /2
for all m ≥ N by the triangle inequality. Thus, the sequence {|an |v } is not equivalent to the null
sequence, i.e., |a|v 6= 0. This proves part i.
Let a, b ∈ Qv and write a = [{an }] and b = [{bn }] for Cauchy sequences {an } and {bn }. Using
|an bn |v = |an |v · |bn |v for all n, we have
[{an }] · [{bn }] = |[{an bn }]|v = [{|an bn |v }] = [{|an |v }] · [{|bn |v }] = |[{an }]|v · |[{bn }]|v .
v
This proves part ii. By the triangle inequality, |an + bn |v − (|an |v + |bn |v ) ≤ 0 for all n. By the
discussion above, this implies that [{|an + bn |v }] 6> [{|an |v }] + [{|bn |v }]. Part iii follows from this
and Lemma 7. The proof of part iv is similar.
Definition: A sequence {an } ⊂ Qv is called Cauchy if for every ε > 0 there is an N ∈ N such
that |am − an |v < ε for all m, n ≥ N .
Technically speaking, the ε here is allowed to be any positive real number. The next lemma
shows that it suffices (if one is so inclined) to only use positive rational ε.
Lemma 8: Suppose a, b ∈ Qv . If a 6= b, then there is a rational number c such that |a − c|v <
|a − b|v .
Proof: As above, write a = [{an }] and b = [{bn }]. Since {an } 6∼ {bn }, {an − bn } is not
equivalent to the null sequence. By Theorem 2, [{an }] − [{bn }]v = |[{an − bn }]|v > 0 so that there
6
is an > 0 and an N ∈ N such that |an − bn |v ≥ for all n ≥ N . Also, for some M ∈ N we have
|an − am |v < /2 for all n, m ≥ M . Let c = aN +M . Then for all n ≥ N + M , |an − c|v < /2 and
|an − bn |v ≥ . In particular, |an − bn |v − |an − c|v ≥ /2 for all n ≥ N + M . By the definitions,
this means that |a − c|v < |b − c|v .
Theorem 3: For all v, the field Qv is topologically complete with respect to the v-adic absolute
value, i.e., every Cauchy sequence in Qv converges.
Proof: Let {rn } be a Cauchy sequence in Qv . If there is an r ∈ Qv and an N ∈ N such that
rn = r for all n ≥ N we are done, since in this case rn → r. So suppose this is not the case. For
each n ∈ N let n0 ∈ N be least such that n0 > n and rn 6= rn0 . For each n ∈ N choose (via Lemma
8) an an ∈ Q such that |rn − an |v < |rn − rn0 |v .
Let > 0. Then there is an N ∈ N such that |rn − rm |v < /3 for all n, m ≥ N . By the
triangle inequality, we have
|an − am |v = |an − rn + rn − rm + rm − am |v
≤ |an − rn |v + |rn − rm |v + |rm − am |v
< |rn − rn0 |v + |rn − rm |v + |rm − rm0 |v
<
for all n, m ≥ N . This shows that {an } is a Cauchy sequence (of rational numbers), i.e., [{an }] ∈ Qv .
Let > 0 again. Then there are is an N ∈ N such that |rn − rm |v , |an − am |v < /3 for all
n, m ≥ N . In particular, |rN − aN |v < /3 and also |aN − [{am }]|v ≤ /3. (Note that aN is viewed
as an element of Qv here.) Using the triangle inequality once more,
|rn − [{am }]|v = rn − rN + rN − aN + aN − [{am }]v
≤ |rn − rN |v + |rN − aN |v + |aN − [{am }]|v
<
for all n ≥ N . Thus, rn → [{am }] ∈ Qv .
Examples:
1. If an ∈ {0, 1, . . . , 9} for all n, then the partial sums for the series
P∞
n=1
an 10−n form a Cauchy
sequence for the archimedean absolute value, so this sum is a (non-negative) real number. In
particular, notice that setting an = 9 for all n gives us a sequence which is equivalent to the
sequence of all 1’s. In other words, “.9 = 1.” This is just a particular case of the following fact: the
7
decimal representation of a real number is not unique.
2. Similar to above, fix a prime p and suppose an ∈ {0, 1, . . . , p − 1} for all n. Then the partial
P∞
sums of n=1 an pn form a Cauchy sequence for the p-adic absolute value, so this sum is an element
of Qp .
Given Theorem 3, one can then proceed to do “calculus” with Qv . Be forewarned that this
gets a little funky for the non-archimedean fields. One can prove the usual results regarding > and
real numbers; this is what really sets R apart from the p-adic numbers Qp .
Similar to the integers Z and their field of quotients Q, inside of the p-adic numbers Qp sit
the p-adic integers Zp . These are the p-adic numbers whose absolute value is no greater than 1:
|a|p ≤ 1. One readily verifies that Zp is a subgroup of Qp (under addition) that is also closed under
multiplication. Moreover, Qp is the field of quotients for Zp .
We know that X 2 − 2 has a root (two, in fact) in R. What are these roots? Technically
speaking, they’re equivalence classes of Cauchy sequences of rational numbers. How could one
produce such a Cauchy sequence? Newton’s method typically works quite well in situations such
as this. Indeed, in this case if we set a1 = 1 and recursively define
an+1 = an +
2 − a2n
2an
√
we get a Cauchy sequence of rational numbers. As a Cauchy sequence in R, it converges to “ 2.”
The same type of construction can sometimes work for p-adic roots as well. The curious can
look up Hensel’s Lemma in just about any text in elementary number theory.
Now that we have the real numbers, we can proceed in general.
Definition: An absolute value on a field F is a function | · | : F → R which satisfies the
following three properties:
i) |a| ≥ 0 with equality if and only if a = 0F ,
ii) |ab| = |a||b| for all a, b ∈ F , and
iii) |a + b| ≤ |a| + |b| for all a, b ∈ F .
If we have
iv) |a + b| ≤ max{|a|, |b|} for all a, b ∈ F
then we say the absolute value is non-archimedean. Otherwise we say it is archimedean.
8
Examples:
1. For a complex number a+bi, a, b ∈ R define |a+bi| =
√
a2 + b2 . This is an archimedean absolute
value on C. This is a special case of extending an absolute value (in this case the archimedean
absolute value on R) to a finite algebraic extension (in this case C).
2. For any field F , define | · | on the field of rational functions F (X) by
A(X) deg(A)−deg(B)
B(X) = e
for A(X), B(X) ∈ F [X]. Any such absolute value is non-archimedean.
3. Let P (X) ∈ F [X] be irreducible over F . Then by unique factorization, any A(X) ∈ F [X]
can be written A = P n Q, where n ≥ 0 and Q(X) ∈ F [X] is not divisible by P . Write ordP (A) = n.
Define | · | on the field of rational functions F (X) by
A(X) ordP (B)−ordP (A)
B(X) = e
for A(X), B(X) ∈ F [X]. Any such absolute value is non-archimedean.
Lemma 9: Suppose F is a field with a non-archimedean absolute value | · |. Then we have
iv’) |a + b| = max{|a|, |b|} for all a, b ∈ F with |a| =
6 |b|.
Proof Note by the definition of absolute value, |1F | = 1. Since (−1F )2 = 1F , we also have
| − 1F | = 1, so that | − a| = |a| for all a ∈ F .
Suppose c ∈ F with |c| < 1. Then
1 = |1F | = |c + 1F − c| ≤ max{|c + 1F |, | − c|} = max{|c + 1F |, |c|}.
This implies that |c + 1F | = 1.
Now suppose a, b ∈ F with |a| < |b|. Then b 6= 0F since |b| > 0. Thus, b has a multiplicative
inverse b−1 . Since 1 = |1F | = |bb−1 | = |b||b−1 |, we have |b|−1 = |b−1 |. Letting c = ab−1 we have
|c| < 1 and by what we’ve shown
|a + b| = |b||c + 1F | = |b| = max{|a|, |b|}.
A corollary of this is
Eisenstein’s Criterion: Suppose P (X) = a0 + a1 X + · · · + an X n ∈ Q[X] and for some prime p we
have |a0 |p = |pan |p ≥ |ai |p . for all i < n. Then P (X) is irreducible over Q.
9
Proof: By considering P ∗ , we may without loss of generality assume that P is a primitive
polynomial in Z[X]. In particular, |an |p = 1 and |a0 |p = p−1 ≥ |ai |p for all i < n. Suppose P
factors over Q. Then it factors over Z, and we may write P (X) = Q(X)R(X) with Q, R primitive
polynomials. Write Q(X) = b0 + · · · + bm X m and R(X) = c0 + · · · + cn−m X n−m . Since a0 = b0 c0 ,
without loss of generality |b0 |p = 1 and |c0 |p = p−1 . Let l be the least index such that |cl | = 1;
such an index exists since R is a primitive polynomial. Then
l
X
|al |p = ci bl−i = 1
i=0
p
since |ci bl−i |p = |ci |p |bl−i |p < 1 for i < l and |cl b0 |p = |cl |p |b0 |p = 1. Since |ai |p < 1 for i < n, we
must have l = n. This gives deg(R) ≥ l = n = deg(P ), so that deg(Q) = 0.
In a manner entirely analogous to what we did with Q, one can topologically complete any
field with an absolute value. For those who are intrigued, Artin’s book Algebraic Numbers and
Algebraic Functions is an excellent starting point.
10