# 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 **Q**, is also true in **R** and all **Q**_{ p}. The terminology is this. **Q** is said to be a global field while all its expansions, **R** included, are said to be local. Thus any relation between a number of rational numbers which is true globally (i.e., in **Q**) is also true locally, i.e., in **R** and all the **Q**_{ p}'s.

The Global-Local Principle asserts a partial converse for the equations involving quadratic forms with integer coefficients: _{ij}x_{i}y_{j} + ∑b_{i}x_{i} + c = 0.*a priori* different) in **R** and, for every prime p, in **Q**_{ p}, then it has a rational solution in **Q**. In other words, a quadratic equation with integer coefficients has a global solution (i.e., in **Q**) if and only if it has solutions in all the local fields (i.e., in **R** and all the **Q**_{ p}.)

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, _{5} = 1_{5} = |α|_{5}|α|_{5} = 1,_{5} = 1,^{k}),*quadratic residue* modulo 5: for none of the residues 0, 1, 2, 3, 4 satisfies **Q** and the reason is that 2 is not a quadratic residue modulo 5!.

### 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 imaginary*i*= √-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 + ... + p^{m} + ...

As we just said, q is well defined as a limit of the sequence {x_{k}} and this is how the identity should be understood. All arithmetic operations are permitted to carry over the limits so that

q | = 1 + p + ... + p^{m} + ... | |

pq | = p + p² + ... + p^{m+1} + ... | |

= 1 + p + ... + p^{m} + ... = 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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