The Local-Global Principle
The Local-Global Principle that was discovered in the 1920s by Helmut Hasse (so it is also known as the Hasse Principle) was the first major discovery that pointed to the utility of p-adic numbers.
The set Q of the rational numbers is a (topological) field which is expanded to either R, the field of the reals, or to the various fields Q p, depending on the norm used. The expansion fields are quite different, but each expands Q algebraically, meaning that, for any pair of rational numbers r and s, the results of the operations
The Global-Local Principle asserts a partial converse for the equations involving quadratic forms with integer coefficients:
As an example, consider the equation x² - 2 = 0. We'll show that this equation has no solution in Q 5. The Local-Global Principle then will imply that the equation has no solution in Q. As a result, √2 is irrational.
First observe, that any solution of x² - 2 = 0 in Q 5 is a 5-adic integer. Indeed,
Note
It is known that every real number has a finite p-adic norm for any prime p: R ⊂Q p. However, as the foregoing discussion shows, the embedding is not algebraic.
rs = t in R does not implyrs = t in Q p, even though,rs = t in Q does.So, x² - 2 = 0 has no solution in Q 5. Curious as it may appear, another well known equation,
x² + 1 = 0, does have a solution in Q 5. In no way this solution relates to the imaginaryi = √-1.
References
- J. R. Goldman, The Queen of Mathematics, A K Peters, 1998
- F. Q. Gouvêa, Local and Global in Number Theory, in The Princeton Companion to Mathematics T. Gowers (ed.), Princeton University Press, 2008
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Copyright © 1996-2018 Alexander BogomolnyStrange as it may appear, we'll write
q = 1 + p + ... + pm + ...
As we just said, q is well defined as a limit of the sequence {xk} and this is how the identity should be understood. All arithmetic operations are permitted to carry over the limits so that
q | = 1 + p + ... + pm + ... | |
pq | = p + p² + ... + pm+1 + ... | |
= 1 + p + ... + pm + ... = q |
So that pq + 1 = q, implying q = 1 / (1 - p). Taking p = 2 we obtain:
-1 = 1 + 2 + 4 + ...
One certainly needs time and effort to get used to such weirdness.
- p-adic Numbers
- p-adic Expansions
- The Local-Global Principle
- Triangulating Squares
- Square Root of 2 is Irrational
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Copyright © 1996-2018 Alexander Bogomolny71472164