Homework 6
Out: 2 Oct. Due: 9 Oct.
Instructions:
1. [Polynomial interpolation]
Consider the set of four points
{
(0
,
1)
,
(1
,
2)
,
(2
,
4)
,
(4
,
2)
}
.
(a) Construct the unique degree3 polynomial (over the reals) that passes through these four points by
writing down and solving a system of linear equations.
(b) Repeat part (a) but using the method of Lagrange interpolation. Show your working.
2. [Polynomials over
GF
(
p
)
]
We have seen that a polynomial
p
(
x
)
of degree
d
over a field
F
has at most
d
roots. However, it may have
fewer than
d
roots.
(a) Fermat’s Little Theorem states that, if
p
is prime, then for any
a
∈ {
1
,
2
, . . . , p

1
}
,
a
p

1
= 1 mod
p
.
Deduce from this that any polynomial over
GF
(
p
)
is equivalent to a polynomial of degree at most
p

1
.
(b) Write down all polynomials over
GF
(2)
that have no roots. Explain your answer.
(c) Write down all polynomials over
GF
(3)
that have no roots. Explain your answer.
(d) Give an example of a polynomial that has at least one root over
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 Fall '08
 HAULLGREN
 Polynomials, Numerical Analysis, Secret sharing, 342G

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