In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, Ostrowski's theorem, due to
Alexander Ostrowski
Alexander Markowich Ostrowski ( uk, Олександр Маркович Островський; russian: Алекса́ндр Ма́ркович Остро́вский; 25 September 1893, in Kiev, Russian Empire – 20 November 1986, in Monta ...
(1916), states that every non-trivial
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
on the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s
is equivalent to either the usual real absolute value or a
-adic absolute value.
Definitions
Raising an
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
to a power less than 1 always results in another absolute value. Two absolute values
and
on a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' are defined to be equivalent if there exists a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
such that
:
The trivial absolute value on any field ''K'' is defined to be
:
The real absolute value on the
rationals
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
is the standard
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
on the reals, defined to be
:
This is sometimes written with a subscript 1 instead of
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
.
For a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, the -adic
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
on
is defined as follows: any non-zero rational can be written uniquely as
, where and are coprime integers not divisible by , and is an integer; so we define
:
Proof
Consider a non-trivial absolute value on the rationals
. We consider two cases:
:
It suffices for us to consider the valuation of integers greater than one. For, if we find
for which
for all naturals greater than one, then this relation trivially holds for 0 and 1, and for positive rationals
:
and for negative rationals
:
Case (1)
This case implies that there exists
such that
By the properties of an absolute value,
and
, so
(it cannot be zero). It therefore follows that .
Now, let
with . Express in
base :
:
Hence
:
so
Then we see, by the properties of an absolute value:
:
As each of the terms in the sum is smaller than
, it follows:
:
Therefore,
:
However, as
, we have
:
which implies
:
Together with the condition
the above argument shows that
regardless of the choice of (otherwise
, implying
). As a result, the initial condition above must be satisfied by any .
Thus for any choice of natural numbers , we get
:
i.e.
:
By symmetry, this inequality is an equality.
Since were arbitrary, there is a constant
for which
, i.e.
for all naturals . As per the above remarks, we easily see that
for all rationals, thus demonstrating equivalence to the real absolute value.
Case (2)
As this valuation is
non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to ...
, there must be a natural number for which
Factoring into
primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
:
:
yields that there exists
such that
We claim that in fact this is so for ''only'' one.
Suppose ''per contra'' that are distinct primes with absolute value less than 1. First, let
be such that
. Since
and
are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
, there are
such that
This yields
:
a contradiction.
So we must have
for some , and
for . Letting
:
we see that for general positive naturals
:
As per the above remarks, we see that
for all rationals, implying that the absolute value is equivalent to the -adic one.
One can also show a stronger conclusion, namely, that
is a nontrivial absolute value
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicond ...
either
for some
or
for some
.
Another Ostrowski's theorem
Another theorem states that any field, complete with respect to an
absolute value (algebra)#Types of absolute value, Archimedean absolute value, is (algebraically and topologically) isomorphic to either the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
or the
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. This is sometimes also referred to as Ostrowski's theorem.
[Cassels (1986) p. 33]
See also
*
Valuation (algebra) In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inh ...
References
*
*
*
*{{cite journal, last=Ostrowski , first=Alexander , authorlink = Alexander Ostrowski, title = Über einige Lösungen der Funktionalgleichung φ(x)·φ(y)=φ(xy) , edition = 2nd, year = 1916, journal = Acta Mathematica, issn = 0001-5962, volume = 41, issue = 1, pages = 271–284, doi = 10.1007/BF02422947, doi-access = free
Theorems in algebraic number theory