POLYNOMIALS WITH A ROOT MOD M

FOR EVERY M BUT NO INTEGER ROOT

Ian Johnson and Alicia Lamarche

Goal• If a polynomial has an integer root, of course it must

have that same root mod m for every . This issue often arises in abstract algebra where we may use the contrapositive form saying that if we can show that no solution exists mod m for some m, then there is no integer solution.• But we should be aware that the converse is false.

That is, when there is no integer root, it may still be possible to have a root mod m for every m. We are interested in this case.

Preliminaries• The Chinese Remainder Theorem

• Hensel’s Lifting Lemma

The Legendre Symbol• Definition

• Working modulo p• Example: Working modulo 5

• A square times a square yields a square.• Example:

• A square times a non-square yields a non-square.• Example:

• A non-square times a non-square yields a square.• Example:

Main Result• We wish to show that has roots modulo m for every m.• The author in the original paper constructs his polynomials by

finding a root modulo p, where r is a quadratic residue (or a square) modulo p.• Then, Hensel's Lifting Lemma implies that quadratic residues modulo

p are also quadratic residues modulo any power of p.• Once we have a root modulo p for all p, using the Chinese Remainder

Theorem, we can put them together to obtain any integer m.• We can be sure that this will include every integer as a consequence

of the Fundamental Theorem of Arithmetic.

• Thus, our new goal is to show that has roots modulo p for every prime p.

Relevant Question• Is there an example of a polynomial with these

properties having degree less than 9?

• Yes, f(x) has a degree of 6. It is also necessary that a and b are not squares in the integers and that and .

Example

• Polynomial Generator