14:332:548, Error Control Coding Homework 3 Rutgers University 1. (a) Show that the polynomial X2 + 1 is irreducible over GF (3). (b) Construct GF (32) using X2 + 1 as an irreducible polynomial...

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14:332:548, Error Control Coding Homework 3 Rutgers University 1. (a) Show that the polynomial X2 + 1 is irreducible over GF (3). (b) Construct GF (32) using X2 + 1 as an irreducible polynomial (describe its addition and multi- plication table). Also, give the power representation and vector representation of each element of GF (32) 2. (a) Show that the polynomial X5 +X3 + 1 is irreducible over GF (2). (b) Let α be a primitive element inGF (24). Use Table 3.3 in the slides to find the roots of f(X) = X3 + α6X2 + α9X + α9. 3. Suppose we constructGF (23) using x3+x2+1 as an irreducible polynomial. Express each nonzero element of GF (23) as a power of a primitive element of GF (23). 4. Let F = GF(2) and suppose that the field Φ = GF ( 24 ) is represented as binary polynmials modulo the irreducible polynomial x4 +x3 +x2 +x+1. Show that in this representation, α+1 is a primitive element in Φ, where α is a root of x4 + x3 + x2 + x+ 1. 5. Let F be a field and Φ be an extension field of F with extension degree [Φ : F ] = h <∞. let β be an element in φ and denote bym the smallest positive integer such that the elements 1, β, β2, . . . , βm are linearly dependent over f . (a) verify that m ≤ h. (b) show that if a(x) is a nonzero polynomial in f [x] such that a(β) = 0, then deg a(x) ≥ m. (c) show that there exists a unique monic polynomial mβ(x) of degree exactly m over f such that mβ(β) = 0. (a monic polynomial is a polynomial in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.) (d) show that the polynomial mβ(x) is irreducible over f . (e) show that if a(x) is a nonzero polynomial in f [x] such that a(β) = 0, then mβ(x) divides a(x). hint: show that β is a root of the remainder polynomial obtained when a(x) is divided by mβ(x). (f) let f = gf (2) and φ = f [ξ]/(ξ3 + ξ + 1). compute the polynomial mξ3(x). hint: identify the representations in f 3 of ξ3i for i = 0, 1, 2, 3 according to the basis ω(1 ξ ξ2). then check the linear dependence of those representations. 1 let="" β="" be="" an="" element="" in="" φ="" and="" denote="" bym="" the="" smallest="" positive="" integer="" such="" that="" the="" elements="" 1,="" β,="" β2,="" .="" .="" .="" ,="" βm="" are="" linearly="" dependent="" over="" f="" .="" (a)="" verify="" that="" m="" ≤="" h.="" (b)="" show="" that="" if="" a(x)="" is="" a="" nonzero="" polynomial="" in="" f="" [x]="" such="" that="" a(β)="0," then="" deg="" a(x)="" ≥="" m.="" (c)="" show="" that="" there="" exists="" a="" unique="" monic="" polynomial="" mβ(x)="" of="" degree="" exactly="" m="" over="" f="" such="" that="" mβ(β)="0." (a="" monic="" polynomial="" is="" a="" polynomial="" in="" which="" the="" leading="" coefficient="" (the="" nonzero="" coefficient="" of="" highest="" degree)="" is="" equal="" to="" 1.)="" (d)="" show="" that="" the="" polynomial="" mβ(x)="" is="" irreducible="" over="" f="" .="" (e)="" show="" that="" if="" a(x)="" is="" a="" nonzero="" polynomial="" in="" f="" [x]="" such="" that="" a(β)="0," then="" mβ(x)="" divides="" a(x).="" hint:="" show="" that="" β="" is="" a="" root="" of="" the="" remainder="" polynomial="" obtained="" when="" a(x)="" is="" divided="" by="" mβ(x).="" (f)="" let="" f="GF" (2)="" and="" φ="F" [ξ]/(ξ3="" +="" ξ="" +="" 1).="" compute="" the="" polynomial="" mξ3(x).="" hint:="" identify="" the="" representations="" in="" f="" 3="" of="" ξ3i="" for="" i="0," 1,="" 2,="" 3="" according="" to="" the="" basis="" ω(1="" ξ="" ξ2).="" then="" check="" the="" linear="" dependence="" of="" those="" representations.="">