Math 328K (Daniel Allcock)
Homework 9, due Friday Apr. 1, 2016
Problem 1. Use Hensel’s lemma to find all solutions to x2 + x + 3 ≡ 0 mod 81.
Problem 2. How many solutions are there to the congruence x3 ≡ 5 mod 65536?
(Hint: you do not need to find all solutions—just say how many there are. State
your reasoning clearly.)
Problem 3. Suppose f (x) is a polynomial with integer coefficients, p is a prime,
and n > 1 is an integer. Suppose rn−1 is a solution of f (x) ≡ 0 mod pn−1 . Assume
also that f 0 (rn−1 ) ≡ 0 mod p, so that f FAILS the derivative test. Then prove
the following “all or nothing” result. Either (i) there is no lift of rn−1 to a root
of f (x) modulo pn ; or (ii) every lift of rn−1 to an element of Zpn is a root of f (x)
modulo pn .
[Which one applies in any given example can be worked out by just picking a
lift of rn−1 to an element of Zpn and checking if it’s a root. If it is, then all of lifts
are; if it’s not then none of them are.]
Problem 4. Find all roots of f (x) = 2x3 + x2 + 7x + 1 mod 25. (Even though this
modulus is small enough to use brute force on, you must use Hensel’s lemma. At
one of the solutions modulo 5, the derivative test fails; you should use the previous
problem to find the lifts of that root.)
Problem 5. For every n ≥ 1, find all solutions to x2 ≡ 1 mod 2n . (Hint: for every
n ≥ 3, there are four solutions. You did the case n = 3 on an earlier homework
assignment.)