a. Show that the polynomial x 2 + 1 is irreducible in Z 3 [x]. b. Let

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a. Show that the polynomial x2 + 1 is irreducible in Z3[x]. 

b. Let a be a zero of x2 + 1 in an extension field of Z3 As in Example 29.19, give the multiplication and addition tables for the nine elements of Z3(α), written in the order 0, 1, 2, α, 2α, 1 + α, 1 + 2α, 2 + α, and 2 + 2α.


Data from in Example 29.19

The polynomial p(x) = x2 + x + 1 in Z2[x] is irreducible over Z2 by Theorem 23.10, since neither element 0 nor element 1 of Z2 is a zero of p(x). By Theorem 29.3, we know that there is an extension field ∈ of Z2 containing a zero α of x2 + x + 1. By Theorem 29.18, Z2(α) has as elements 0 + 0α, 1 + 0α, 0 + 1α, and 1 + 1α, that is, 0, 1, α, and 1 + α. This gives us a new finite field, of four elements! The addition and multiplication tables for this field are shown in Tables 29.20 and 29.21. For example, to compute (1 + α)(l + α) in Z2(α), we observe that since p(α) = α2 + α + 1 = 0, then α2 = - α - 1 = α + 1. 

Therefore, (1 + α)(1 +α)= 1 + α + α+ α2 = 1 + α= 1 + α + 1 = α.

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