In
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
, the Grunwald–Wang theorem is a
local-global principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each ...
stating that—except in some precisely defined cases—an element ''x'' in a
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
''K'' is an ''n''th power in ''K'' if it is an ''n''th power in the
completion for all but finitely many primes
of ''K''. For example, a
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 ration ...
is a square of a rational number if it is a square of a
''p''-adic number for almost all primes ''p''. The Grunwald–Wang theorem is an example of a
local-global principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each ...
.
It was introduced by , but there was a mistake in this original version that was found and corrected by . The theorem considered by Grunwald and Wang was more general than the one stated above as they discussed the existence of cyclic extensions with certain local properties, and the statement about ''n''th powers is a consequence of this.
History
, a student of
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...
, gave an incorrect proof of the erroneous statement that an element in a number field is an ''n''th power if it is an ''n''th power locally almost everywhere. gave another incorrect proof of this incorrect statement. However discovered the following counter-example: 16 is a ''p''-adic 8th power for all odd primes ''p'', but is not a rational or 2-adic 8th power. In his doctoral thesis written under
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
, Wang gave and proved the correct formulation of Grunwald's assertion, by describing the rare cases when it fails. This result is what is now known as the Grunwald–Wang theorem. The history of Wang's counterexample is discussed by
Wang's counter-example
Grunwald's original claim that an element that is an ''n''th power almost everywhere locally is an ''n''th power globally can fail in two distinct ways: the element can be an ''n''th power almost everywhere locally but not everywhere locally, or it can be an ''n''th power everywhere locally but not globally.
An element that is an ''n''th power almost everywhere locally but not everywhere locally
The element 16 in the rationals is an 8th power at all places except 2, but is not an 8th power in the 2-adic numbers.
It is clear that 16 is not a 2-adic 8th power, and hence not a rational 8th power, since the 2-adic valuation of 16 is 4 which is not divisible by 8.
Generally, 16 is an 8th power in a field ''K'' if and only if the polynomial
has a root in ''K''. Write
:
Thus, 16 is an 8th power in ''K'' if and only if 2, −2 or −1 is a square in ''K''. Let ''p'' be any odd prime. It follows from the multiplicativity of the
Legendre symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residu ...
that 2, −2 or −1 is a square modulo ''p''. Hence, by
Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
, 2, −2 or −1 is a square in
.
An element that is an ''n''th power everywhere locally but not globally
16 is not an 8th power in
although it is an 8th power locally everywhere (i.e. in
for all ''p''). This follows from the above and the equality
.
A consequence of Wang's counter-example
Wang's counterexample has the following interesting consequence showing that one cannot always find a cyclic Galois extension of a given degree of a number field in which finitely many given prime places split in a specified way:
There exists no cyclic degree 8 extension
in which the prime 2 is totally inert (i.e., such that
is unramified of degree 8).
Special fields
For any
let
:
Note that the
th
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
is
:
A field is called ''s-special'' if it contains
, but neither
,
nor
.
Statement of the theorem
Consider a number field ''K'' and a natural number ''n''. Let ''S'' be a finite (possibly empty) set of primes of ''K'' and put
:
The Grunwald–Wang theorem says that
:
unless we are in the ''special case'' which occurs when the following two conditions both hold:
#
is ''s''-special with an
such that
divides ''n''.
#
contains the ''special set''
consisting of those (necessarily 2-adic) primes
such that
is ''s''-special.
In the special case the failure of the Hasse principle is finite of order 2: the kernel of
:
is Z/2Z, generated by the element η.
Explanation of Wang's counter-example
The field of rational numbers
is 2-special since it contains
, but neither
,
nor
. The special set is
. Thus, the special case in the Grunwald–Wang theorem occurs when ''n'' is divisible by 8, and ''S'' contains 2. This explains Wang's counter-example and shows that it is minimal. It is also seen that an element in
is an ''n''th power if it is a ''p''-adic ''n''th power for all ''p''.
The field
is 2-special as well, but with
. This explains the other counter-example above.
[See Chapter X of Artin–Tate.]
See also
*The
Hasse norm theorem Hasse is both a surname and a given name. Notable people with the name include:
Surname:
* Clara H. Hasse (1880–1926), American botanist
* Helmut Hasse (1898–1979), German mathematician
* Henry Hasse (1913–1977), US writer of science ficti ...
states that for cyclic extensions an element is a norm if it is a norm everywhere locally.
Notes
References
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{{DEFAULTSORT:Grunwald-Wang theorem
Class field theory
Theorems in algebraic number theory