Math 342
Homework Assignment #6 (Due Friday, April 8, 5PM in my office)
1. Which of the following polynomials in F [x] are irreducible?
(a) f (x) = x2 + x + 1, F = Z2
(b) f (x) = x2 + x + 1, F = GF (4)
(c) f (x) = x4 + x + 1, F = Z2
(d) f (x) = ax3 + bx2 + ax + 1, F = GF (4)
2. Find the gcd of 2x5 + x4 + 2x3 + x2 + 1 and 2x4 + x3 + x + 1 in Z3 [x]
3. Show that for any finite field GF (q) and any nonzero element α ∈ GF (q), αq−1 = 1.
4. A primitive element of a finite field GF (q) is an element α ∈ GF (q) such that
{α, α2 , . . . , αq−1 } = GF (q) \ {0}. It is a fact that every finite field has a primitive
element.
Find all primitive elements of each of the fields Z3 , GF (4) and Z7 .
5. Recall from class that the polynomials f (x) = x3 + x + 1 and g(x) = x3 + x2 + 1 are
both irreducible over Z2 . So, both Rf := Z2 [x]/f (x) and Rg := Z2 [x]/g(x) are fields.
(a) Find |Rf | and |Rg |.
(b) Write out the addition and multiplication tables for both fields Rf and Rg .
(c) Show that these two fields are isomorphic. That is, find a bijection k : Rf → Rg
such that for all α, β ∈ Rf such that k(α + β) = k(α) + k(β) and k(α · β) =
k(α) · k(β).
6. Find all primitive elements of GF (8). Use the representation:
GF (8) = Z2 [x]/(x3 + x + 1).
7. Let {a1 , . . . , a7 } denote the 7 non-zero elements of
over GF (8) defined by the parity check matrix

1 1 1 1 1

H = a1 a2 a3 a4 a5
a21 a22 a23 a24 a25
GF (8). Let C be the linear code

1 1
a6 a7 
a26 a27
(a) Find the dimension and minimum distance of C.
(b) How many cosets does C have?
(c) Find 51 coset leaders and their corresponding syndromes.
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