U.C. Berkeley — CS70: Discrete Mathematics and Probability
Problem Set 6
Lecturers: Umesh Vazirani & Christos H. Papadimitriou
Due October 6, 2006 at 4:00pm
Problem Set 6 Solutions
1.
Polynomials with no roots
A polynomial
p
(
x
) of degree
n
over a field
F
has at most
n
roots. But it does not need to have
n
roots,
nor it must have roots at all.
(a) Write all polynomials in
GF
2
having no roots over
GF
2
.
(b) Write all polynomials in
GF
3
having no roots over
GF
3
.
(c) Find a polynomial
p
(
x
) which has roots over
GF
3
, but not over
Z
.
Solution
: Recall that polynomials of degree
≥
q
in
GF
q
can be rewritten as polynomials of degree
≤
q
as
x
q
= 1 for all
x
∈
GF
q
.
(a) The constant 1 polynomial is the only polynomial without roots. All nonconstant polynomials
have degree 1 and hence have one root.
(b) These can be found by iterating through all 3
3
polynomials, but it is not necessary. First, notice
that all nonzero constant polynomials have no roots and that all degree 1 polynomials have roots.
Then, we are left with the 2
*
3
*
3 = 18 polynomials of degree 2. Here, observe that those with 0
constant term will have 0 as a root and exclude them. Now, you have 2
*
3
*
2 = 12 polynomials
to consider. Moreover, you can arrange these in pairs
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 Fall '08
 HAULLGREN
 Polynomials, Numerical Analysis

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