Primes And Polynomials
A result on the prime divisors of polynomial values
|
| Cropped from source |
Issai Schur was a mathematician who obtained his doctorate over a hundred years ago. He was a student of the great group theorist Ferdinand Frobenius. Schur worked in various areas and proved many deep results, including some theorems in basic number theory.
Today we discuss a nice lemma due to Schur. Actually, it’s a theorem.
There are many things named after Schur. Wikipedia has compiled a list of them:
Frobenius-Schur indicator Herz-Schur multiplier Jordan-Schur theorem Lehmer-Schur algorithm Schur algebra Schur class Schur complement method Schur complement Schur decomposition Schur functor Schur index Schur multiplier Schur number Schur orthogonality relations Schur polynomial Schur product theorem Schur test Schur-convex function Schur-Horn theorem Schur-Weyl duality Schur-Zassenhaus theorem Schur’s inequality Schur’s lemma (from Riemannian geometry) Schur’s lemma Schur’s property Schur’s theorem
This lists two Schur lemmas. But when you click on Wikipedia’s “Schur’s lemma” page, there are three Schur lemmas. No, wait—there are four, including one called “Schur’s test.” But the result we are interested in is classed as a theorem. Wikipedia’s single page for “Schur’s theorem” lists not just five but six Schur’s theorems. This is one higher than the count eventually reached in Monty Python’s famous “Spanish Inquisition” skit. We want the sixth one, which is quite useful—like a lemma.
The Result
Here is Schur’s lemma—no, theorem—published in 1912.
Theorem 1 Let
be a non-constant polynomial with integer coefficients. Then
has infinitely many prime divisors.
Here 
is fine. As is 
Schur’s theorem is quite fun, even just to prove. Here is a nice version from a paper by Ram Murty, whose work on extensions of Euclid’s famous proof of the infinitude of primes we covered last summer. We follow Murty’s proof.
Proof: Suppose 
where 
Now 
for some integer 
This proof was simple but clever. Here is a concrete version for the polynomial 
where
For some 
An Application
As the proof hints, Schur’s theorem is a proper extension of Euclid’s, which is just the case 
Theorem 2 Suppose that
is a perfect square for all integers
large enough. Then
for some polynomial
.
There are many proofs of this theorem. One uses the famous Hilbert Irreducibility theorem, which we also discussed last summer. Another proof uses Schur’s lemma and the fact that if 
Proof: We can factor 
By a similar token, since 
![{\mathbb{Z}[x]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbb%7BZ%7D%5Bx%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
Open Problems
We did mention Schur’s long list of things named after him. Did people name things after others more often years ago? Or is naming them still common-place?
[fixed last proof]
I’m confused… why is it that p^2 dividing h(n) implies p divides h'(n)? This obviously resembles the fact that if f^2 divides h then f divides h’, but with actual numbers plugged in. It’s not obvious why it would still work in that context; because while you can differentiate h, the polynomial, you can’t different h(n), the number…
Thanks—you are right, the shortcut was incorrect.
The patern and shapes of prime Numbers might to follow the polynmial equations and theirs with complex roots with mod(T) that is moviment of the time.the flux of the time is Given by the velocities of the sequênces measured by the imaginary numbers
I don’t see Schur rings in the list. When I began as a DPhil student in Oxford in 1968, my first job was to read Wielandt’s book on Finite Permutation Groups: one of its five chapters was on “The method of Schur”, i.e. Schur rings: these are subrings of the group ring spanned by sums of subsets of the group, these subsets forming a partition. (Wielandt was a student of Schur.)