Number system extending the rational numbers
In this article, unless otherwise stated,
p
denotes a prime number that is fixed once for all.
In
number theory
, given a
prime number
p
, the
p
-adic numbers
form an extension of the
rational numbers
which is distinct from the
real numbers
, though with some similar properties;
p
-adic numbers can be written in a form similar to (possibly
infinite
)
decimals
, but with digits based on a
prime number
p
rather than ten, and extending to the left rather than to the right.
For example, comparing the expansion of the rational number
in
base
3
vs. the
3
-adic expansion,
Formally, given a prime number
p
, a
p
-adic number can be defined as a
series
where
k
is an
integer
(possibly negative), and each
is an integer such that
A
p
-adic integer
is a
p
-adic number such that
In general the series that represents a
p
-adic number is not
convergent
in the usual sense, but it is convergent for the
p
-adic absolute value
where
k
is the least integer
i
such that
(if all
are zero, one has the zero
p
-adic number, which has
0
as its
p
-adic absolute value).
Every rational number can be uniquely expressed as the sum of a series as above, with respect to the
p
-adic absolute value. This allows considering rational numbers as special
p
-adic numbers, and alternatively defining the
p
-adic numbers as the
completion
of the rational numbers for the
p
-adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
p
-adic numbers were first described by
Kurt Hensel
in 1897,
[1]
though, with hindsight, some of
Ernst Kummer's
earlier work can be interpreted as implicitly using
p
-adic numbers.
[note 1]
Motivation
[
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]
Roughly speaking,
modular arithmetic
modulo a positive integer
n
consists of "approximating" every integer by the remainder of its
division
by
n
, called its
residue modulo
n
. The main property of modular arithmetic is that the residue modulo
n
of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo
n
. If one knows that the absolute value of the result is less than
n/2
, this allows a computation of the result which does not involve any integer larger than
n
.
For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the
Chinese remainder theorem
for recovering the result modulo the product of the moduli.
Another method discovered by
Kurt Hensel
consists of using a prime modulus
p
, and applying
Hensel's lemma
for recovering iteratively the result modulo
If the process is continued infinitely, this provides eventually a result which is a
p
-adic number.
Basic lemmas
[
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]
The theory of
p
-adic numbers is fundamentally based on the two following lemmas
Every nonzero rational number can be written
where
v
,
m
, and
n
are integers and neither
m
nor
n
is divisible by
p
.
The exponent
v
is uniquely determined by the rational number and is called its
p
-adic valuation
(this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from the
fundamental theorem of arithmetic
.
Every nonzero rational number
r
of valuation
v
can be uniquely written
where
s
is a rational number of valuation greater than
v
, and
a
is an integer such that
The proof of this lemma results from
modular arithmetic
: By the above lemma,
where
m
and
n
are integers
coprime
with
p
. The
modular inverse
of
n
is an integer
q
such that
for some integer
h
. Therefore, one has
and
The
Euclidean division
of
by
p
gives
where
since
mq
is not divisible by
p
. So,
which is the desired result.
This can be iterated starting from
s
instead of
r
, giving the following.
Given a nonzero rational number
r
of valuation
v
and a positive integer
k
, there are a rational number
of nonnegative valuation and
k
uniquely defined nonnegative integers
less than
p
such that
and
The
p
-adic numbers are essentially obtained by continuing this infinitely to produce an
infinite series
.
p
-adic series
[
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]
The
p
-adic numbers are commonly defined by means of
p
-adic series.
A
p
-adic series
is a
formal power series
of the form
where
is an integer and the
are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of
is not divisible by
p
).
Every rational number may be viewed as a
p
-adic series with a single nonzero term, consisting of its factorization of the form
with
n
and
d
both coprime with
p
.
Two
p
-adic series
and
are
equivalent
if there is an integer
N
such that, for every integer
the rational number
is zero or has a
p
-adic valuation greater than
n
.
A
p
-adic series
is
normalized
if either all
are integers such that
and
or all
are zero. In the latter case, the series is called the
zero series
.
Every
p
-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see
§ Normalization of a
p
-adic series
, below.
In other words, the equivalence of
p
-adic series is an
equivalence relation
, and each
equivalence class
contains exactly one normalized
p
-adic series.
The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of
p
-adic series. That is, denoting the equivalence with
~
, if
S
,
T
and
U
are nonzero
p
-adic series such that
one has
The
p
-adic numbers are often defined as the equivalence classes of
p
-adic series, in a similar way as the definition of the real numbers as equivalence classes of
Cauchy sequences
. The uniqueness property of normalization, allows uniquely representing any
p
-adic number by the corresponding normalized
p
-adic series. The compatibility of the series equivalence leads almost immediately to basic properties of
p
-adic numbers:
- Addition
,
multiplication
and
multiplicative inverse
of
p
-adic numbers are defined as for
formal power series
, followed by the normalization of the result.
- With these operations, the
p
-adic numbers form a
field
, which is an
extension field
of the rational numbers.
- The
valuation
of a nonzero
p
-adic number
x
, commonly denoted
is the exponent of
p
in the first non zero term of the corresponding normalized series; the valuation of zero is
- The
p
-adic absolute value
of a nonzero
p
-adic number
x
, is
for the zero
p
-adic number, one has
Normalization of a
p
-adic series
[
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]
Starting with the series
the first above lemma allows getting an equivalent series such that the
p
-adic valuation of
is zero. For that, one considers the first nonzero
If its
p
-adic valuation is zero, it suffices to change
v
into
i
, that is to start the summation from
v
. Otherwise, the
p
-adic valuation of
is
and
where the valuation of
is zero; so, one gets an equivalent series by changing
to
0
and
to
Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of
is zero.
Then, if the series is not normalized, consider the first nonzero
that is not an integer in the interval
The second above lemma allows writing it
one gets n equivalent series by replacing
with
and adding
to
Iterating this process, possibly infinitely many times, provides eventually the desired normalized
p
-adic series.
Definition
[
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]
There are several equivalent definitions of
p
-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use
completion
of a
discrete valuation ring
(see
§ p-adic integers
),
completion of a metric space
(see
§ Topological properties
), or
inverse limits
(see
§ Modular properties
).
A
p
-adic number can be defined as a
normalized
p
-adic series
. Since there are other equivalent definitions that are commonly used, one says often that a normalized
p
-adic series
represents
a
p
-adic number, instead of saying that it
is
a
p
-adic number.
One can say also that any
p
-adic series represents a
p
-adic number, since every
p
-adic series is equivalent to a unique normalized
p
-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of
p
-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations on
p
-adic numbers, since the series operations are compatible with equivalence of
p
-adic series.
With these operations,
p
-adic numbers form a
field
called the
field of
p
-adic numbers
and denoted
or
There is a unique
field homomorphism
from the rational numbers into the
p
-adic numbers, which maps a rational number to its
p
-adic expansion. The
image
of this homomorphism is commonly identified with the field of rational numbers. This allows considering the
p
-adic numbers as an
extension field
of the rational numbers, and the rational numbers as a
subfield
of the
p
-adic numbers.
The
valuation
of a nonzero
p
-adic number
x
, commonly denoted
is the exponent of
p
in the first nonzero term of every
p
-adic series that represents
x
. By convention,
that is, the valuation of zero is
This valuation is a
discrete valuation
. The restriction of this valuation to the rational numbers is the
p
-adic valuation of
that is, the exponent
v
in the factorization of a rational number as
with both
n
and
d
coprime
with
p
.
p
-adic integers
[
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]
The
p
-adic integers
are the
p
-adic numbers with a nonnegative valuation.
A
p
-adic integer can be represented as a sequence
of residues
x
e
mod
p
e
for each integer
e
, satisfying the compatibility relations
for
i < j
.
Every
integer
is a
p
-adic integer (including zero, since
). The rational numbers of the form
with
d
coprime with
p
and
are also
p
-adic integers (for the reason that
d
has an inverse mod
p
e
for every
e
).
The
p
-adic integers form a
commutative ring
, denoted
or
, that has the following properties.
- It is an
integral domain
, since it is a
subring
of a field, or since the first term of the series representation of the product of two non zero
p
-adic series is the product of their first terms.
- The
units
(invertible elements) of
are the
p
-adic numbers of valuation zero.
- It is a
principal ideal domain
, such that each
ideal
is generated by a power of
p
.
- It is a
local ring
of
Krull dimension
one, since its only
prime ideals
are the
zero ideal
and the ideal generated by
p
, the unique
maximal ideal
.
- It is a
discrete valuation ring
, since this results from the preceding properties.
- It is the
completion
of the local ring
which is the
localization
of
at the prime ideal
The last property provides a definition of the
p
-adic numbers that is equivalent to the above one: the field of the
p
-adic numbers is the
field of fractions
of the completion of the localization of the integers at the prime ideal generated by
p
.
Topological properties
[
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]
The
p
-adic valuation allows defining an
absolute value
on
p
-adic numbers: the
p
-adic absolute value of a nonzero
p
-adic number
x
is
where
is the
p
-adic valuation of
x
. The
p
-adic absolute value of
is
This is an absolute value that satisfies the
strong triangle inequality
since, for every
x
and
y
one has
- if and only if
Moreover, if
one has
This makes the
p
-adic numbers a
metric space
, and even an
ultrametric space
, with the
p
-adic distance defined by
As a metric space, the
p
-adic numbers form the
completion
of the rational numbers equipped with the
p
-adic absolute value. This provides another way for defining the
p
-adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every
Cauchy sequence
a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the
partial sums
of a
p
-adic series, and thus a unique normalized
p
-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized
p
-adic series instead of equivalence classes of Cauchy sequences).
As the metric is defined from a discrete valuation, every
open ball
is also
closed
. More precisely, the open ball
equals the closed ball
where
v
is the least integer such that
Similarly,
where
w
is the greatest integer such that
This implies that the
p
-adic numbers form a
locally compact space
, and the
p
-adic integers?that is, the ball
?form a
compact space
.
p
-adic expansion of rational numbers
[
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]
The
decimal expansion
of a positive
rational number
is its representation as a
series
where
is an integer and each
is also an
integer
such that
This expansion can be computed by
long division
of the numerator by the denominator, which is itself based on the following theorem: If
is a rational number such that
there is an integer
such that
and
with
The decimal expansion is obtained by repeatedly applying this result to the remainder
which in the iteration assumes the role of the original rational number
.
The
p
-
adic expansion
of a rational number is defined similarly, but with a different division step. More precisely, given a fixed
prime number
, every nonzero rational number
can be uniquely written as
where
is a (possibly negative) integer,
and
are
coprime integers
both coprime with
, and
is positive. The integer
is the
p
-adic valuation
of
, denoted
and
is its
p
-adic absolute value
, denoted
(the absolute value is small when the valuation is large). The division step consists of writing
where
is an integer such that
and
is either zero, or a rational number such that
(that is,
).
The
-
adic expansion
of
is the
formal power series
obtained by repeating indefinitely the
above
division step on successive remainders. In a
p
-adic expansion, all
are integers such that
If
with
, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of
in
base-
p
.
The existence and the computation of the
p
-adic expansion of a rational number results from
Bezout's identity
in the following way. If, as above,
and
and
are coprime, there exist integers
and
such that
So
Then, the
Euclidean division
of
by
gives
with
This gives the division step as
so that in the iteration
is the new rational number.
The uniqueness of the division step and of the whole
p
-adic expansion is easy: if
one has
This means
divides
Since
and
the following must be true:
and
Thus, one gets
and since
divides
it must be that
The
p
-adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a
convergent series
with the
p
-adic absolute value.
In the standard
p
-adic notation, the digits are written in the same order as in a
standard base-
p
system
, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side.
The
p
-adic expansion of a rational number is eventually
periodic
.
Conversely
, a series
with
converges (for the
p
-adic absolute value) to a rational number
if and only if
it is eventually periodic; in this case, the series is the
p
-adic expansion of that rational number. The
proof
is similar to that of the similar result for
repeating decimals
.
Example
[
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]
Let us compute the 5-adic expansion of
Bezout's identity for 5 and the denominator 3 is
(for larger examples, this can be computed with the
extended Euclidean algorithm
). Thus
For the next step, one has to expand
(the factor 5 has to be viewed as a "
shift
" of the
p
-adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded). To expand
, we start from the same Bezout's identity and multiply it by
, giving
The "integer part"
is not in the right interval. So, one has to use
Euclidean division
by
for getting
giving
and the expansion in the first step becomes
Similarly, one has
and
As the "remainder"
has already been found, the process can be continued easily, giving coefficients
for
odd
powers of five, and
for
even
powers.
Or in the standard 5-adic notation
with the
ellipsis
on the left hand side.
Positional notation
[
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]
It is possible to use a
positional notation
similar to that which is used to represent numbers in
base
p
.
Let
be a normalized
p
-adic series, i.e. each
is an integer in the interval
One can suppose that
by setting
for
(if
), and adding the resulting zero terms to the series.
If
the positional notation consists of writing the
consecutively, ordered by decreasing values of
i
, often with
p
appearing on the right as an index:
So, the computation of the
example above
shows that
and
When
a separating dot is added before the digits with negative index, and, if the index
p
is present, it appears just after the separating dot. For example,
and
If a
p
-adic representation is finite on the left (that is,
for large values of
i
), then it has the value of a nonnegative rational number of the form
with
integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in
base
p
. For these rational numbers, the two representations are the same.
Modular properties
[
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]
The
quotient ring
may be identified with the
ring
of the integers
modulo
This can be shown by remarking that every
p
-adic integer, represented by its normalized
p
-adic series, is congruent modulo
with its
partial sum
whose value is an integer in the interval
A straightforward verification shows that this defines a
ring isomorphism
from
to
The
inverse limit
of the rings
is defined as the ring formed by the sequences
such that
and
for every
i
.
The mapping that maps a normalized
p
-adic series to the sequence of its partial sums is a ring isomorphism from
to the inverse limit of the
This provides another way for defining
p
-adic integers (
up to
an isomorphism).
This definition of
p
-adic integers is specially useful for practical computations, as allowing building
p
-adic integers by successive approximations.
For example, for computing the
p
-adic (multiplicative) inverse of an integer, one can use
Newton's method
, starting from the inverse modulo
p
; then, each Newton step computes the inverse modulo
from the inverse modulo
The same method can be used for computing the
p
-adic
square root
of an integer that is a
quadratic residue
modulo
p
. This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in
. Applying Newton's method to find the square root requires
to be larger than twice the given integer, which is quickly satisfied.
Hensel lifting
is a similar method that allows to "lift" the factorization modulo
p
of a polynomial with integer coefficients to a factorization modulo
for large values of
n
. This is commonly used by
polynomial factorization
algorithms.
Notation
[
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]
There are several different conventions for writing
p
-adic expansions. So far this article has used a notation for
p
-adic expansions in which
powers
of
p
increase from right to left. With this right-to-left notation the 3-adic expansion of
for example, is written as
When performing arithmetic in this notation, digits are
carried
to the left. It is also possible to write
p
-adic expansions so that the powers of
p
increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of
is
p
-adic expansions may be written with
other sets of digits
instead of
{0, 1, ...,
p
???1
}. For example, the
3
-adic expansion of
can be written using
balanced ternary
digits
{
1
, 0, 1
}, with
1
representing negative one, as
In fact any set of
p
integers which are in distinct
residue classes
modulo
p
may be used as
p
-adic digits. In number theory,
Teichmuller representatives
are sometimes used as digits.
[2]
Quote notation
is a variant of the
p
-adic representation of
rational numbers
that was proposed in 1979 by
Eric Hehner
and
Nigel Horspool
for implementing on computers the (exact) arithmetic with these numbers.
[3]
Cardinality
[
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]
Both
and
are
uncountable
and have the
cardinality of the continuum
.
[4]
For
this results from the
p
-adic representation, which defines a
bijection
of
on the
power set
For
this results from its expression as a
countably infinite
union
of copies of
:
Algebraic closure
[
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]
contains
and is a field of
characteristic
0
.
Because
0
can be written as sum of squares,
[5]
cannot be turned into an
ordered field
.
The field of
real numbers
has only a single proper
algebraic extension
: the
complex numbers
. In other words, this
quadratic extension
is already
algebraically closed
. By contrast, the
algebraic closure
of
, denoted
has infinite degree,
[6]
that is,
has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the
p
-adic valuation to
the latter is not (metrically) complete.
[7]
[8]
Its (metric) completion is called
or
.
[8]
[9]
Here an end is reached, as
is algebraically closed.
[8]
[10]
However unlike
this field is not
locally compact
.
[9]
and
are isomorphic as rings,
[11]
so we may regard
as
endowed with an exotic metric. The proof of existence of such a field isomorphism relies on the
axiom of choice
, and does not provide an explicit example of such an isomorphism (that is, it is not
constructive
).
If
is any finite
Galois extension
of
, the
Galois group
is
solvable
. Thus, the Galois group
is
prosolvable
.
Multiplicative group
[
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]
contains the
n
-th
cyclotomic field
(
n
> 2
) if and only if
n
|
p
? 1
.
[12]
For instance, the
n
-th cyclotomic field is a subfield of
if and only if
n
= 1, 2, 3, 4, 6
, or
12
. In particular, there is no multiplicative
p
-
torsion
in
if
p
> 2
. Also,
?1
is the only non-trivial torsion element in
.
Given a
natural number
k
, the
index
of the multiplicative group of the
k
-th powers of the non-zero elements of
in
is finite.
The number
e
, defined as the sum of
reciprocals
of
factorials
, is not a member of any
p
-adic field; but
for
. For
p
= 2
one must take at least the fourth power.
[13]
(Thus a number with similar properties as
e
? namely a
p
-th root of
e
p
? is a member of
for all
p
.)
Local?global principle
[
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]
Helmut Hasse
's
local?global principle
is said to hold for an equation if it can be solved over the rational numbers
if and only if
it can be solved over the real numbers and over the
p
-adic numbers for every prime
p
. This principle holds, for example, for equations given by
quadratic forms
, but fails for higher polynomials in several indeterminates.
Rational arithmetic with Hensel lifting
[
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]
Generalizations and related concepts
[
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]
The reals and the
p
-adic numbers are the completions of the rationals; it is also possible to complete other fields, for instance general
algebraic number fields
, in an analogous way. This will be described now.
Suppose
D
is a
Dedekind domain
and
E
is its
field of fractions
. Pick a non-zero
prime ideal
P
of
D
. If
x
is a non-zero element of
E
, then
xD
is a
fractional ideal
and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of
D
. We write ord
P
(
x
) for the exponent of
P
in this factorization, and for any choice of number
c
greater than 1 we can set
Completing with respect to this absolute value
|?|
P
yields a field
E
P
, the proper generalization of the field of
p
-adic numbers to this setting. The choice of
c
does not change the completion (different choices yield the same concept of Cauchy sequence, so the same completion). It is convenient, when the
residue field
D
/
P
is finite, to take for
c
the size of
D
/
P
.
For example, when
E
is a
number field
,
Ostrowski's theorem
says that every non-trivial
non-Archimedean absolute value
on
E
arises as some
|?|
P
. The remaining non-trivial absolute values on
E
arise from the different embeddings of
E
into the real or complex numbers. (In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of
E
into the fields
C
p
, thus putting the description of all
the non-trivial absolute values of a number field on a common footing.)
Often, one needs to simultaneously keep track of all the above-mentioned completions when
E
is a number field (or more generally a
global field
), which are seen as encoding "local" information. This is accomplished by
adele rings
and
idele groups
.
p
-adic integers can be extended to
p
-adic solenoids
. There is a map from
to the
circle group
whose fibers are the
p
-adic integers
, in analogy to how there is a map from
to the circle whose fibers are
.
See also
[
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]
Notes
[
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]
Citations
[
edit
]
- ^
(
Hensel 1897
)
- ^
(
Hazewinkel 2009
, p. 342)
- ^
(
Hehner & Horspool 1979
, pp. 124?134)
- ^
(
Robert 2000
, Chapter 1 Section 1.1)
- ^
According to
Hensel's lemma
contains a square root of
?7
, so that
and if
p
> 2
then also by Hensel's lemma
contains a square root of
1 ?
p
, thus
- ^
(
Gouvea 1997
, Corollary 5.3.10)
- ^
(
Gouvea 1997
, Theorem 5.7.4)
- ^
a
b
c
(
Cassels 1986
, p. 149)
- ^
a
b
(
Koblitz 1980
, p. 13)
- ^
(
Gouvea 1997
, Proposition 5.7.8)
- ^
Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example Lang’s
Algebra
X §1), and both
and
have characteristic zero and the cardinality of the continuum.
- ^
(
Gouvea 1997
, Proposition 3.4.2)
- ^
(
Robert 2000
, Section 4.1)
References
[
edit
]
- Cassels, J. W. S.
(1986),
Local Fields
, London Mathematical Society Student Texts, vol. 3,
Cambridge University Press
,
ISBN
0-521-31525-5
,
Zbl
0595.12006
- Dedekind, Richard
;
Weber, Heinrich
(2012),
Theory of Algebraic Functions of One Variable
, History of mathematics, vol. 39, American Mathematical Society,
ISBN
978-0-8218-8330-3
. — Translation into English by
John Stillwell
of
Theorie der algebraischen Functionen einer Veranderlichen
(1882).
- Gouvea, F. Q. (March 1994), "A Marvelous Proof",
American Mathematical Monthly
,
101
(3): 203?222,
doi
:
10.2307/2975598
,
JSTOR
2975598
- Gouvea, Fernando Q. (1997),
p
-adic Numbers: An Introduction
(2nd ed.), Springer,
ISBN
3-540-62911-4
,
Zbl
0874.11002
- Hazewinkel, M., ed. (2009),
Handbook of Algebra
, vol. 6, North Holland, p. 342,
ISBN
978-0-444-53257-2
- Hehner, Eric C. R.
; Horspool, R. Nigel (1979),
"A new representation of the rational numbers for fast easy arithmetic"
,
SIAM Journal on Computing
,
8
(2): 124?134,
CiteSeerX
10.1.1.64.7714
,
doi
:
10.1137/0208011
- Hensel, Kurt
(1897),
"Uber eine neue Begrundung der Theorie der algebraischen Zahlen"
,
Jahresbericht der Deutschen Mathematiker-Vereinigung
,
6
(3): 83?88
- Kelley, John L.
(2008) [1955],
General Topology
, New York: Ishi Press,
ISBN
978-0-923891-55-8
- Koblitz, Neal
(1980),
p
-adic analysis: a short course on recent work
, London Mathematical Society Lecture Note Series, vol. 46,
Cambridge University Press
,
ISBN
0-521-28060-5
,
Zbl
0439.12011
- Robert, Alain M. (2000),
A Course in
p
-adic Analysis
, Springer,
ISBN
0-387-98669-3
Further reading
[
edit
]
- Bachman, George (1964),
Introduction to
p
-adic Numbers and Valuation Theory
, Academic Press,
ISBN
0-12-070268-1
- Borevich, Z. I.
;
Shafarevich, I. R.
(1986),
Number Theory
, Pure and Applied Mathematics, vol. 20, Boston, MA: Academic Press,
ISBN
978-0-12-117851-2
,
MR
0195803
- Koblitz, Neal
(1984),
p
-adic Numbers,
p
-adic Analysis, and Zeta-Functions
,
Graduate Texts in Mathematics
, vol. 58 (2nd ed.), Springer,
ISBN
0-387-96017-1
- Mahler, Kurt
(1981),
p
-adic numbers and their functions
, Cambridge Tracts in Mathematics, vol. 76 (2nd ed.), Cambridge:
Cambridge University Press
,
ISBN
0-521-23102-7
,
Zbl
0444.12013
- Steen, Lynn Arthur
(1978),
Counterexamples in Topology
, Dover,
ISBN
0-486-68735-X
External links
[
edit
]