Polynomials with a root m od m for e very m but n o i nteger root

1. Ian Johnson and Alicia Lamarche Polynomials with a root mod mfor every m but no integer root

2. 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.

3. Preliminaries • The Chinese Remainder Theorem • Hensel’s Lifting Lemma

4. 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:

5. 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'sLifting 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.

6. 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 .

7. Example • Polynomial Generator