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Problem 1.
Construct the multiplication table of the
group G given
by the presentation:
( r, s | r4 = s2 = 1,
sr = r3s )
You may assume that the distinct group elements
are:
1, r, r2,r3, s, rs,
r2s, r3s
Also compute in GF(2)G
the product
(1+r3+r2s)(r+s+r3s)
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Problem 2.
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.
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Problem 3.
Let R
be a commutative ring with non-zero elements a
and b
such that
ab = 0
Prove that R
is not a field.
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Problem 4.
Let ksi
be the primitive element of GF(26)
defined by ksi = x mod x6
+ x + 1. Compute the orders of the elements
of ksii
for i=0,1, ... , 62.
Summarize your results in a log/order table. For which i's
are the ksii's
primitive? Do you see a pattern? Make a conjecture about this
pattern.
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Problem 5.
Compute the addition and multiplication tables
for the ring
R3 = GF(2)[x]/(x3+1)
Also express each of the following ideals in
the ring R3 as
a set of elements of R3 .
(0), (1+x), (x2+x+1), (1), (x2+1), (x3+1),
(x5+x+1).
For example,
(0) = { 0 } and ( x4 + x2
+1 ) = { 0, x2 + x+1 }