Homework 5
Due: Tuesday, October 28, 1997
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Problem 1. The
polynomial
p(x) = x4 + x3 +
x2 + x + 1
is irreducible over GF(2),
and therefore the algebra of polynomials modulo p(x)
is. GF(24).
Let
ksi = x mod
p(x).
Show that ksi
is not a primitive element, and therefore p(x)
is not a primitive polynomial. Show that
alpha = 1 + ksi
is primitive and find its minimum
polynomial, which is a primitive polynomial.
Problem 2. Prove
that GF(7) is
a field.
Problem 3. Consider
GF(25)
represented as
GF(2)[x] mod x5
+ x2 + 1,
where p(x)
= x5 + x2 + 1 is
a primitive polynomial. Let
ksi = x mod x5
+ x2 + 1
Find all the roots of p(x).
Find all roots of the dual polynomial p*(x).
Last Modified: October 15, 1997