EEM 465
FUNDAMENTALS of DATA COMMUNICATIONS
Lecture 9
Lecturer
Assist.Prof.Dr. Nuray At
2
Minimal Polynomials
Let
. The minimal polynomial of 𝛼 with respect to GF(q) is the
smallest degree nonzero polynomial p(x) in GF(q)[x] such that
The principal design question:
If we want polynomial p(x) with coefficients in GF(q) to have a root 𝛼 from
GF(qm), what other roots must the polynomial have?
Conjugacy Classes
Let
. The conjugates of
with respect to GF(q) are the elements
Conjugates of 𝛼 with respect to GF(q) form a set called the conjugacy class of 𝛼
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Example: Find conjugacy classes of
respect to GF(2).
q=2, m=3,
, 𝛼 a primitive element, with
GF(8)=
{0}
{1}
Theorem: The conjugacy class of
elements, where
and d|m.
with respect to GF(q) contains d
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Example:
a) Let 𝛼 be an element of order 3 in GF(16). The conjugates of 𝛼 wrt GF(2)
b)
Let 𝛼 be an element of order 6 in GF(25). The conjugacy class of 𝛼 wrt GF(5)
c)
Let 𝛼 be an element of order 63 in GF(64). The conjugacy class of 𝛼 wrt GF(4)
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Example: Conjugacy classes of
p=2, m=4,
{0}
{1}
wrt GF(2), 𝛼 a primitive element
Theorem: Let
. Let p(x) be the minimal polynomial of 𝛼 wrt GF(q).
The roots of p(x) are exactly the conjugates of 𝛼 wrt GF(q).
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Example: The minimal polynomials of the elements in GF(8) wrt GF(2).
Recall the construction of GF(8).
Conjugacy class
Associated minimal polynomial
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Factorization of xn-1 (n=qm-1)
Theorem: The set of nonzero elements in GF(qm) form the complete set of roots
of the expression
roots of unity.
. Equivalently, the elements of GF(qm) are
The minimal polynomials wrt GF(q) of the nonzero elements in GF(qm) thus
provide the complete factorization of
into irreducible polynomials.
Example: The factorization of
in GF(2)[x] :
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Factorization of xn-1 (n ≠ qm-1)
If n| qm-1, then there is an element of order n in GF(qm). Let
order n in GF(qm).
 By definition,
 Also the elements
are distinct.
be an element of
Therefore, n roots of xn-1 are the n consecutive powers of
Look for smallest extension field GF(qm) of GF(q) so that n|qm -1
Once the desired primitive root has been located, form the conjugacy classes and
compute the associated minimal polynomials.
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Example: Factor
using polynomials with coefficients from GF(2).
5|10, 11 is not power of a prime
5|15, 16 = 24 . Roots of
lie in GF(16)
Conjugacy classes of GF(16) wrt GF(2):
Let
be a root of
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Cyclotomic Cosets:
In many cases the common base of the elements in a conjugacy class is obvious
within the context. When the base element is deleted, the resulting partition of
powers of the base element form cyclotomic cosets
Example:
Conjugacy class
Cyclotomic cosets