Local-global result for when an element in a number field is an nth power
In algebraic number theory, the Grunwald–Wang theorem is a local-global principle stating that—except in some precisely defined cases—an element "x" in a number field "K" is an "n"th power in "K" if it is an "n"th power in the completion formula_1 for all but finitely many primes formula_2 of "K". For example, a rational number is a square of a rational number if it is a square of a "p"-adic number for almost all primes "p". 
It was introduced by Wilhelm Grunwald (1933), but there was a mistake in this original version that was found and corrected by Shianghao Wang (1948). 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.
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Some days later I was with Artin in his office when Wang appeared. He said he had a counterexample to a lemma which had been used in the proof. An hour or two later, he produced a counterexample to the theorem itself... Of course he [Artin] was astonished, as were all of us students, that a famous theorem with two published proofs, one of which we had all heard in the seminar without our noticing anything, could be wrong. 
John Tate, quoted by Peter Roquette (2005, p.30)
, a student of Helmut Hasse, 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. George Whaples (1942) 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, 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 Peter Roquette (2005, section 5.3)
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 formula_3 has a root in "K". Write
formula_4
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 that 2, −2 or −1 is a square modulo "p". Hence, by Hensel's lemma, 2, −2 or −1 is a square in formula_5.
An element that is an "n"th power everywhere locally but not globally.
16 is not an 8th power in formula_6 although it is an 8th power locally everywhere (i.e. in formula_7 for all "p"). This follows from the above and the equality formula_8.
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 formula_9 in which the prime 2 is totally inert (i.e., such that formula_10 is unramified of degree 8).
Special fields.
For any formula_11 let
formula_12
Note that the formula_13th cyclotomic field is
formula_14
A field is called "s-special" if it contains formula_15, but neither formula_16, formula_17 nor formula_18.
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
formula_19
The Grunwald–Wang theorem says that
formula_20
unless we are in the "special case" which occurs when the following two conditions both hold:
In the special case the failure of the Hasse principle is finite of order 2: the kernel of
 formula_28
is Z/2Z, generated by the element η.
Explanation of Wang's counter-example.
The field of rational numbers formula_29 is 2-special since it contains formula_30, but neither formula_16, formula_32 nor formula_33. The special set is formula_34. 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 formula_35 is an "n"th power if it is a "p"-adic "n"th power for all "p".
The field formula_36 is 2-special as well, but with formula_37. This explains the other counter-example above.
Notes.
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