Square roots have no unexpected linear relationships

IMO medalist Iurie Boreico has an article in an issue of the Harvard College Mathematics Review about his favorite problem:

Let Image may be NSFW.
Clik here to view. be distinct squarefree integers. Show that if Image may be NSFW.
Clik here to view. are not all zero, then the sum Image may be NSFW.
Clik here to view. is nonzero.

In other words, the problem is to show that there are no “unexpected” linear relationships between the fractions Image may be NSFW.
Clik here to view.. This is one of those nice problems that justifies an intuition without being amenable to the obvious attacks. Iurie gives several ingenious Olympiad-style arguments, i.e. arguments that don’t require much machinery, including a clearly motivated argument that serves as a good way to motivate the definition of a Galois group.

Today I want to share a different argument that generalizes a simple argument for special cases, but it has the unfortunate property that in full generality it is severe overkill.

First of all, it may not be clear that the problem statement implies that numbers such as Image may be NSFW.
Clik here to view. are irrational; it only implies that they are non-integer. However, it’s known that the sum of algebraic integers is again an algebraic integer, and it is also known, via the rational root theorem, that the only rational algebraic integers are the integers. (Even if you aren’t familiar with the proofs of these results, I would like to stress that they are straightforward compared to the results we’ll be citing by the end of this post!) So the problem statement does involve showing that a large class of sums that we expect to be irrational are in fact irrational.

The very simplest special case of this problem requires showing that Image may be NSFW.
Clik here to view. is irrational for Image may be NSFW.
Clik here to view. a square-free integer. This is a well-known classical problem in number theory dating back to the Greeks; one can argue by contradiction and prime factorization, from the rational root theorem, or from the fact that Image may be NSFW.
Clik here to view. has a periodic continued fraction. The first argument is elementary, the first and second generalize readily to Image may be NSFW.
Clik here to view. powers, but the third is my favorite; it is the basis of a certain proof you may have seen of the irrationality of Image may be NSFW.
Clik here to view. which disguises a proof of the identity

Image may be NSFW.
Clik here to view..

There is a fourth argument I would like to present, and it may seem silly at first: Image may be NSFW.
Clik here to view. cannot be an integer because Image may be NSFW.
Clik here to view. is not a quadratic residue Image may be NSFW.
Clik here to view..

Why is this sufficient? Suppose Image may be NSFW.
Clik here to view. for some integer Image may be NSFW.
Clik here to view.. Then Image may be NSFW.
Clik here to view., which implies that Image may be NSFW.
Clik here to view., but checking Image may be NSFW.
Clik here to view. we readily see that this can’t occur; contradiction.

This is the seed of an important idea, which can be stated roughly as follows.

Important idea: Certain statements have the property that if they are true locally (i.e. modulo a prime), then they are true for the integers.

More precisely, we need the following result.

Proposition: Let Image may be NSFW.
Clik here to view. be a monic polynomial with integer coefficients. If Image may be NSFW.
Clik here to view. is irreducible over Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view. is irreducible over Image may be NSFW.
Clik here to view.. (The reverse implication does not hold in general.)

In other words, if an algebraic integer is “locally irrational,” then it must be irrational. This is the first conceptual ingredient of our proof.

Here’s the second. Suppose I want to prove that Image may be NSFW.
Clik here to view. is irrational. I want to reduce this to a statement about the irrationality of a single square root, since I already know how to deal with that by finding a suitable prime (under the assumption that such a prime exists). So here’s the trick: find a prime such that Image may be NSFW.
Clik here to view. is a quadratic residue, but Image may be NSFW.
Clik here to view. isn’t!

Here’s how this works with Image may be NSFW.
Clik here to view.. Suppose Image may be NSFW.
Clik here to view. for some integer Image may be NSFW.
Clik here to view.. Since Image may be NSFW.
Clik here to view., this is equivalent to Image may be NSFW.
Clik here to view., but again checking Image may be NSFW.
Clik here to view. (or using Euler’s criterion) we can readily verify that Image may be NSFW.
Clik here to view. isn’t a quadratic residue Image may be NSFW.
Clik here to view.; contradiction.

Problems of this sort were often given on my abstract algebra problem sets before we covered Galois theory. I think the intended solution was either to square (which, as Iurie mentions, stops working for sums of more than four square roots) or to show that the minimal polynomial is irreducible. But one of the easiest ways to show that a polynomial is irreducible is to show that it’s irreducible modulo a prime – this is the motivation behind Eisenstein’s criterion – and so we are led back to this idea anyway, which we can implement without actually computing the minimal polynomial.

On another problem set we were asked to show that Image may be NSFW.
Clik here to view. was not an element of Image may be NSFW.
Clik here to view.; since Image may be NSFW.
Clik here to view. is an algebraic integer, this is equivalent to showing that Image may be NSFW.
Clik here to view. isn’t of the form Image may be NSFW.
Clik here to view. for integers Image may be NSFW.
Clik here to view.. (Well, up to the small problem that I don’t have a good proof that the ring of integers in Image may be NSFW.
Clik here to view. is Image may be NSFW.
Clik here to view., but for now I’ll take this as given.) The trick is to find a prime relative to which Image may be NSFW.
Clik here to view. is a cubic residue, but Image may be NSFW.
Clik here to view. isn’t. It turns out that Image may be NSFW.
Clik here to view. is such a prime, since Image may be NSFW.
Clik here to view. but

Image may be NSFW.
Clik here to view.

(this is the cubic generalization of Euler’s criterion). So this “find a prime” method works well for specific cases, but I haven’t described a general way to find the magical primes we need. The intuition we want here is that numbers that are unrelated multiplicatively behave independently modulo various primes. So we need to prove a statement like the following.

Conjecture: Let Image may be NSFW.
Clik here to view. be distinct primes. Then there exists an arbitrarily large prime Image may be NSFW.
Clik here to view. relative to which Image may be NSFW.
Clik here to view. are quadratic residues, but Image may be NSFW.
Clik here to view. isn’t.

First let’s verify that this conjecture implies the desired result. Given a sum Image may be NSFW.
Clik here to view. of integer combinations of distinct squarefree integers that might be equal to zero, we’re going to induct on the number of distinct primes Image may be NSFW.
Clik here to view. that appears in Image may be NSFW.
Clik here to view.. Clearly the original problem statement is satisfied for Image may be NSFW.
Clik here to view.. Now, any such sum can be written as a polynomial in some set Image may be NSFW.
Clik here to view. of square roots of primes. The monomials of this polynomial are squarefree, i.e. the polynomial is linear in each variable, so we can write

Image may be NSFW.
Clik here to view.

where Image may be NSFW.
Clik here to view. are integer linear combinations of products of distinct elements of the set Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. is nonzero by the inductive hypothesis. Actually, the inductive hypothesis gives us something stronger, which is that all of the conjugates of Image may be NSFW.
Clik here to view. are also nonzero, hence the minimal polynomial of Image may be NSFW.
Clik here to view. has nonzero constant term, i.e. the norm of Image may be NSFW.
Clik here to view. is nonzero.

This is important. Now we pick a prime Image may be NSFW.
Clik here to view. satisfying the hypotheses of the conjecture such that Image may be NSFW.
Clik here to view. does not divide this constant term (which is why we need Image may be NSFW.
Clik here to view. to be arbitrarily large). Then any “reduction” of Image may be NSFW.
Clik here to view. is still nonzero and Image may be NSFW.
Clik here to view. is a contradiction.

The hard part, of course, is proving the conjecture. I see no way to do it that doesn’t require the following two tools: quadratic reciprocity and Dirichlet’s theorem. Quadratic reciprocity is a gem of number theory; it not only validates the assumption that numbers behave independently modulo primes, but it makes the astonishing claim that whether a prime Image may be NSFW.
Clik here to view. is a quadratic residue Image may be NSFW.
Clik here to view. depends only on the residue class of Image may be NSFW.
Clik here to view.. Dirichlet’s theorem is another validation of intuition: it lets us find arbitrarily large primes in arbitrary residue classes exactly like the ones we need for quadratic reciprocity.

Now for every Image may be NSFW.
Clik here to view. there exists a residue class Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. is a quadratic residue relative to primes in this residue class by quadratic reciprocity. There also exists a residue class Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. is not a quadratic residue relative to primes in this residue class, again by quadratic reciprocity. If we choose all the Image may be NSFW.
Clik here to view. to be congruent Image may be NSFW.
Clik here to view. (and we can always do this), then by CRT there exists a residue class Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. are quadratic residues relative to primes in this residue class but Image may be NSFW.
Clik here to view. isn’t. By Dirichlet’s theorem, this residue class contains arbitrarily large primes, and we’re done.

If that seemed like more work than was necessary to solve this problem, it probably was. But I think this argument is a good demonstration of the local perspective in number theory. It also suggests that it would be useful to understand higher-order reciprocity laws in order to generalize the proof to higher powers, but this leads into waters so deep they justify themselves as of independent interest.